Linear Algebra : Exercises

LINEAR ALGEBRA Exercises Tereza Kovářová Translation of the Czech Version of Exercises by Petra Vondráková Ostrava 2024 Preface Linear Algebra (470-2205/02) Material for seminars and individual practice This work is a collection of exercises without worked-out solutions, but with correct results provided. These exercises are intended to help students of Linear Algebra individually test their acquired knowledge and practice their skills in the subject. Exercises are thematically divided into sections corresponding to the topics covered in lectures and seminars of the specified course. This is the English version of the Czech title ”Sb´ırka neˇreˇsen´ych pˇr´ıklad˚u z line´arn´ı algebry”by Petra Vondr´akov´a (ˇ Sarmanov´a), 2009, translated and written by Tereza Kov´aˇrov´a, 2024. 1 Contents 1 Vectors 3 2 Matrices 4 3 Systems of Linear Equations 6 4 Matrix Inverse 8 5 Vector Spaces 10 6 Linear Dependence, Linear Combination, Basis 12 7 Vector Coordinates, Rank of Matrix, Frobeni’s Theorem 14 8 Linear Mappings 15 9 Bilinear Forms 17 10 Quadratic Forms 19 11 Scalar Product, Gram-Schmidt Process 21 12 Determinants 22 2 1 Vectors 1. List all the elements of the set A1→A2→A3→A4,where A1={↑1,0,1}, A1={1,2}, A1={2,3}, A1={0}. 2. Does the set N5contain the element [0,1,↑1,2,1] or the element [2,4,6,8]? By Nwe understand the set of all natural numbers. Explain your answer. 3. Are the vectors uand vfrom R4the same, if u=[ln ↓e, cos ω 2,tan ω 4,sin ω 3] and v= [sin ω 6,log 1,cot ω 4,cos ω 6 ]? Explain your answer. 4. Is the vector [↓↑2,e,tan ω 2,↑1,ln(↑3)] an element of R5?Explainyouranswer. 5. Evaluate: 2 ↓3!"5tanω 4↑ln e2,↑sin ω 3,cos 5 6ω#↑2 cos 5 3ω,1 2 sin(→pi 2),&log 104↑4 tan(→ω 4) 8 . 6. For each u,v↔Rnand ε,ϑ↔Rprove that the following holds: (a) u+v=v+u (b) ε(u+v)=εu+εv (c) (ε+ϑ)u=εu+ϑu 7. Find and also graph the vector that is the result of 1 2[↑4,↑2] + 2[1,2]. 8. Compute the scalar product (the dot product), the length, and the angle of the vectors aand bfrom R3,wherea=[ ↓2,↑↓2,0] and b=[3,0,0]. 9. Find at least one vector orthogonal to v=[3,-1,2]. Solutions 1. A1→A2→A3→A4={[↑1,1,2,0],[↑1,1,3,0],[↑1,2,2,0],[↑1,2,3,0],[0,1,2,0],[0,1,3,0], [0,2,2,0],[0,2,3,0],[1,1,2,0],[1,1,3,0],[1,2,2,0],[1,2,3,0]}. 2. No. 3. Yes. 4. No. 5. [1,0,↑3]. 8. a·b=3 ↓2, |a|=2,|b|=3,ϖ=ω 4. 9. For instance, [↑1,↑1,1]. 3 2 Matrices 1. Given the matrices A=  12 2 21↑2 2↑21 and B=  011 ↑12↑1 1↑20 , find the matrix C, such that C=2A↑3B. 2. Given the matrix Aand the vector v: A=  120 202 ↑111  ,v=  ↑1 1 0 , compute Av. 3. Given the matrices Aand B,where A=  110 ↑11 2 20↑1 and B=  10 ↑22 01  , compute AB. 4. Given the matrix A,where A=  3↑12 4↑32 213  , compute A2=AA. 5. Find the matrix Xso that the given equation is satisfied. )1234 ↑1↑2↑3↑4*X=)2413 ↑2↑4↑1↑3*. 6. Find the matrix Xso that the given equation is satisfied. X  114 222 333  =  333 114 222  . 7. Find the matrix Xso that the given equation is satisfied. X  111 111 111  =  333 444 222  . 8. With what matrix do we need to multiply (from the right) another 3 →3 matrix to switch its 3-rd column with the 2-nd column? 4 9. With what matrix do we need to multiply (from the left) another 4 →4 matrix to switch its 3-rd row with the 1-st row? 10. Prove that for any matrix Aand any real numbers rand sthe following holds: (a) (r+s)A=rA+sA, (b) r(sA)=(rs)A. Solutions 1. 2A↑3B=  21 1 7↑4↑1 12 2  . 2. Av =  1 ↑2 2 . 3. AB =  ↑12 ↑34 2↑1 . 4. AA =  9210 478 16 ↑215  . 5. X=    0010 1000 0001 0100     . 6. X=  001 100 010  . 7. X=  300 040 002  . 8. X=  100 001 010  . 9. X=    0010 0100 1000 0001     . 5 3 Systems of Linear Equations 1. Use the Gaussian Elimination Method (row reduction) to solve the system of linear equations: x1+2x2+4x3=5 2x1↑x2+3x3=5 ↑x1+3x2+x3=0 . 2. Use the Gaussian Elimination Method (row reduction) to solve the system of linear equations: x1+2x2+4x3=3 2x1↑x2+3x3=3 ↑x1+3x2+x3=2 . 3. Use the Gaussian Elimination Method (row reduction) to find a) all solutions of the linear system 6x1↑9x2+7x3+10x4=3 2x1↑3x2↑3x3+4x4=1 2x1↑3x2+13x3+18x4=1 , b) solution for x2=1. 4. Use the Gaussian Elimination Method (row reduction) to find a) all solutions of the linear system x1+2x2↑x3↑x4=2 2x1↑5x2+x3+3x4=1 4x1↑x2↑x3+2x4=5 , b) solution for x2=1. 5. Find all real solutions of the linear system 2x1+3x2↑x3+x4=1 8x1+12x2↑9x3+8x4=3 4x1+6x2+3x3↑2x4=3 2x1+3x2+9x3↑7x4=3. 6. Find all complex solutions of the linear system x1+i·x2↑2x3=10 x1↑x2+2i·x3=10 x1+3i·x2↑(1 + i)·x3=30. 7. Use the Gaussian Elimination Method to find all the values of aand bfor which the following linear system has infinit number of solutions. Use parameter pto describe those solutions. ax1+2x3=2 5x1+2x2=1 x1↑2x2+bx3=3 . 6 Solutions 1. [x1,x 2,x 3] = [3 ↑2p, 1↑p, p],p↔R 2. No solutions. 3. a) [x1,x 2,x 3,x 4]=[ 1+3p 2,p,0,0],p↔R;b)[x1,x 2,x 3,x 4]=[2,1,0,0] 4. a) [x1,x 2,x 3,x 4]=[1+p, p, ↑1+3p, 0],p↔R;b) [x1,x 2,x 3,x 4]=[2,1,2,0] 5. [x1,x 2,x 3,x 4]=[ 3 5↑3 2s↑1 10 t, s, 1 5+4 5t, t],s,t↔R 6. [x1,x 2,x 3]=[2↑16i, 4↑12i, 2↑6i] 7. a=3,b=4,[x1,x 2,x 3]=[ 2→2p 3,→7+10p 6,p],p↔R 7 4 Matrix Inverse 1. Find the inverse A→1of the matrix A. A=  12 2 21↑2 2↑21 . 2. Find the inverse A→1of the matrix A. A=  011 ↑12↑1 1↑20 . 3. Find the inverse A→1of the matrix A. A=  110 ↑11 2 20↑1 . 4. Let A=  3↑12 4↑32 213  . (a) Find the inverse A→1. (b) Use A→1to find the solution of the given linear system: 3x1↑x2+2x3=7 4x1↑3x2+2x3=4 2x1+x2+3x3=13 5. Use the inverse of the coe!cient matrix to solve the system. 2x1+2x2+x3=↑1 3x1+x2+5x3=↑1 3x1+2x2+3x3=↑1 6. Simplify and evaluate the matrix expression (without computing C→1): C→1Av +C→1Bv =?,where A=  111 010 111  ,B=  101 211 ↑110  ,C=  120 202 ↑111  ,v=  ↑1 1 0 . 7. With what matrix and from which side we multiply another matrix to perform the operation r1↑r2+5r3↗r2. 8 8 Linear Mappings 1. Show that the mapping A:R3↗R3is (is not) linear, if a) A([x1,x 2,x 3]) = [2x2+x3,x 1+x3,3x1+x2↑x3], b) A([x1,x 2,x 3]) = [x1+x2,x 1↑x3,x 2 1]. 2. Show that the mapping A:R3↗R2is (is not) linear, if A([x1,x 2,x 3]) = [2x1+5x2↑x3,↑x1↑2x3↑2]. 3. Show that the mapping A:P2↗R3is (is not) linear, if A(ax2+bx +c)=[a+b, a ↑b, ↑c]. 4. A linear mapping A:R2↗R2is given by the following relations A([1,↑1]) = [2,3],A([↑1,↑1]) = [1,2]. Find the image of [5,1], i.e., A([5,1]) =?. 5. A linear mapping A:P2↗P2is given by A(p)=p(↑x)+p(x+1). a) Show that the mapping is linear. b) Find the image of the polynomial p(x)=2↑5x+6x2. 6. A linear mapping A:R3↗R2is given by relations: A([1,2,0]) = [2,3],A([1,1,1]) = [0,1],A([↑1,3,↑1]) = [1,4]. a) Find the image of [12,2,↑14], i.e., A([12,2,↑14]) =?. b) Find the preimage of [7,3], i.e., A(u) = [7,3], where u=?. 7. A linear mapping A:P2↗R2is given by relations: A(x2+x) = [1,↑1],A(x2+x+1)=[↑1,2],A(x)=[0,↑1]. a) Find the image of the polynomial ↑2x2+3x↑4. b) Find the preimage of the vector [3,↑2] ↔R2. 8. A linear mapping A:R3↗R3is given by A([x1,x 2,x 3]) = (x1↑x2,x 1+x2,x 1+x2+x3). Find a) the null space of Aand its dimension, i.e., N(A) =? and dim(N(A)) =? b) the range of Aand its dimension, i.e., H(A) =? and dim(H(A)) =? 9. A linear mapping A:R3↗R2is given by relations: A([1,1,↑1]) = [2,1],A([1,↑1,1]) = [2,↑2],A([↑1,1,1]) = [0,↑3]. Find the null space of Aand its dimension, i.e., N(A) =? and dim(N(A)) =?. 15 10. A linear transformation A:R3↗R3is given by relations: A([1,1,0]) = [1,↑1,1],A([1,1,1]) = [1,0,1],A([0,1,0]) = [0,↑1,0]. Find a) the null space of Aand its dimension, i.e., N(A) =? and dim(N(A)) =? b) the range of Aand its dimension, i.e., H(A) =? and dim(H(A)) =? 11. A linear mapping A:R3↗R2is given by A([x1,x 2,x 3]) = [x1+2x2,3x2+4x3]. Find the matrix of the mapping Arelative to the bases Eand F,whereEis the standard basis in R3and F=(f1,f2)isabasisinR2,f1=[1,2], f2=[2,2]. 12. A linear mapping A:R4↗R3is given by relations: A([1,1,↑1,0]) = [0,0,0], A([1,2,↑1,↑2]) = [↑1,↑3,1], A([1,0,0,↑1]) = [0,0,0], A([1,1,1,1]) = [5,8,2]. Find the matrix of the mapping Arelative to the standard bases of R4and R3. Solutions 1. a) Yes. b) No. 2. No. 3. Yes. 4. [1,0] 5. a) Yes. b) 5 + 12x+12x2 6. a) [43,19] b) [0,0,↑4] 7. a) [6,↑15] b) 1 + 5x+5x2+t(1 + 3x+2x2) 8. a) N(A)={o},dimN(A)=0 b)H(A)=⇒[1,1,1],[↑1,1,1],[0,0,1]⇑,dimH(A)=3 9. N(A)=⇒[↑1,3,↑1]⇑, dim(N(A)=1 10. a) N(A)=⇒[0,1,1]⇑,dimN(A)=1 b)H(A)=⇒[1,0,1],[0,↑1,0]⇑,dimH(A)=2 11. [A]EF =)↑11 4 11 2↑2*. 12. [A]EF =  1121 2132 0110  . 16 9 Bilinear Forms 1. Decide if the mapping B:P2→P2↗Rgiven by the equality in a) or b) for every p, q ↔P2is a bilinear form. (a) B(p, q)=4p(0)q(1) + p(2)q(3) (b) B(p, q)=p(1)q(↑1) + p(0)q2(1) 2. The bilinear form B:P2→P2↗Ris given by the equality B(p(x),q(x)) = p(0)q(0) + p(1)q(1),where p(x),q(x)↔P2. Find the matrix of this bilinear form relative to the standard basis of P2. 3. A bilinear form B:P2→P2↗Ris given by B(p, q)=p(2)q(3). Find the matrix of Brelative to the basis F=(p1,p 2,p 3), where p1(x)=1,p2(x)=1↑x, p3(x)=(1↑x)2. 4. A bilinear form B:P2→P2↗Ris given by B(p, q)=p(0)q(3), where p, q ↔P2. (a) Find the matrix [B]Fof Brelative to the basis F=(p1,p 2,p 3), where p1(x)=1, p2(x)=x↑1, p3(x)=(x↑1)2. (b) Find the coordinates of p(x)=x+1 and q(x)=x2+1 relative to the basis F=(p1,p 2,p 3). (c) Use the matrix [B]Fto find the value of B(p, q) for polynomials p(x)=x+1 and q(x)=x2+1. 5. A bilinear form B:R3→R3↗Ris defined by (a) Determine its symmetric and its antisymmetric part. (b) Write the matrix of Brelative to the standrd basis, [B]E. (c) Also write the matrices of its symmetric and antisymmetric part, [BS]Eand [BA]E. 6. Decompose the matrix Aof a bilinear form A=  362 ↑203 415   to the sum of its symmetric and antisymmetric part. 17 Solutions 1. (a) yes, (b) no, 2. [B]=  211 111 111   3. [B]F=  1↑24 ↑12↑4 1↑24   4. (a) [B]F=  124 ↑1↑2↑4 124   (b) [p]F=[2,1,0], [q]F=[2,2,1], (c) B(p, q)=10, 5. (a) BS(x, y)=x1y1↑x1y2+3 2x1y3↑x2y1↑1 2x2y3+3 2x3y1↑1 2x3y2, BA(x, y)=↑x1y2+3 2x1y3+x2y1↑1 2x2y3↑3 2x3y1+1 2x3y2, (b) [B]E=  1↑23 00↑1 00 0  ,[BS]E=  1↑13 2 ↑10↑1 2 3 2↑1 20 ,[BA]E=  0↑13 2 10↑1 2 ↑3 2 1 20  6. [B]=  323 202 325  +  04↑1 ↑40 1 1↑10  . 18 10 Quadratic Forms 1. A bilinear form B:R2→R2↗Ris given by B(x, y)=x1y1+3x1y2+5x2y1↑3x2y2. Write the quadratic form QB(x)associatedwiththegivenbilinearformBand its matrix [Q]E 2. A bilinear form B:P2→P2↗Ris given by B(p(x),q(x)) = p(↑1)q(2) + p(2)q(1). (a) Find the matrix of the bilinear form relative to the standard basis E=(x2,x,1) and then find the matrices of its symmetric and antisymmetric part. (b) Find the image of the pair of two polynomials p(x)=↑3x2+5x+2andq(x)=4x2↑2x in the given bilinear form B. (c) Suppose Q(p(x)) is the quadratic form on P2given by Q(p(x)) = 8a2+2c2+4ab+10ac+ 4bc,wherep(x)=ax2+bx+c.IsQ(p(x)) the quadratic form associated with the bilinear form Bgiven above? Justify your answer. 3. Classify the following quadratic forms (Determine their type.): (a) Q:R2↗R,Q(x)=3x2 2↑2x1x2, (b) Q:R3↗R,Q(x)=↑3x2 1↑4x2 2↑2x2 3+6x1x2+2x1x3↑4x2x3, (c) Q:R3↗R,Q(x)=x2 1+x2 2+x2 3↑4x1x2↑4x2x3, (d) Q:R3↗R,Q(x)=x2 1+4x2 2+8x2 3↑4x1x2+6x1x3↑12x2x3, (e) Q:R3↗R,Q(x)=x2 1+x2 2+4x2 3↑x1x2↑x2x3. 4. If a quadratic form Q(x) is given, find basis Tin which the matrix of the form is diagonal, [Q]T=D. Classify the quadratic form. (a) Q:R2↗R,Q(x)=3x2 1+2x1x2, (b) Q:R2↗R,Q(x)=↑2x2 1+2x1x2↑3x2 2, (c) Q:R2↗R,Q(x)=↑2x2 1↑2x1x2↑1 2x2 2, (d) Q:R3↗R,Q(x)=2x2 1+4x2 2+29x2 3+4x1x2↑12x1x3↑4x2x3. 5. Find the diagonal matrix D, that is congruent to the matrix A: A=  2↑10 ↑12↑1 0↑12 . Solutions 1. Q(x)=x2 1+8x1x2↑3x2 2, [Q]=)14 4↑3*19 2. (a) [B]=  865 ↑20↑1 532 ,[BS]=  825 201 512 ,[BA]=  040 ↑40↑2 020  (b) B(p, q)=↑62, (c) Yes. 3. (a) indefinite (b) negatively definite (c) indefinite (d) indefinite (e) positively definite 4. (a) indefinite, basis T=([1,0],[1,↑3]), [Q]T=D=)30 0↑3*. (b) negatively definite, basis T=([1,0],[1,2]), [Q]T=D=)↑20 0↑10 *. (c) negatively semidefinite, basis T=([1,0],[1,↑2]), [Q]T=D=)↑20 00 *. (d) positively definite, basis T=((1,0,0),(↑1,1,0),(5,↑2,1)), [Q]T=D=  200 020 003  . 5. For instance D=  20 0 06 0 0012  . 20 11 Scalar Product, Gram-Schmidt Process 1. Consider the vector space R2with u,v↔R2. Decide wheather the next expressions define a scalar product (u,v)onR2. (a) (u,v)=4u1v1+2u1v2+2u2v1+3u2v2, (b) (u,v)=2u2 1+4u2 2+6u2 3↑u1v2↑2u2v3. 2. Suppose, on the vector space R3,ascalarproductisdefinedas(x,y)=3x1y1+4x2y2+x3y3. Use basis F=(f1,f2,f3), where f1=[0,0,1], f2=[0,1,1], f3=[1,2,1], to find a basis Bthat is orthogonal with respect to the given scalar product. 3. Use Gram-Schmidt process to find an orthonormal basis of the span <u1,u2,u3>,where u1=[1,1,1,1], u2=[0,1,↑1,0], u3=[1,3,↑1,3], and the scalar product is given by (u,v)=u1v1+u2v2+u3v3+u4v4. 4. Consider the vector space R3with the scalar product defined as (u,v)=u1v1+u2v2+u3v3. Find the coordinates of the vector x=[3,4,5] relative to the orthogonal basis E=-[1,↑1,1],[1,1,0],[↑1 2,1 2,1].. Use the orthogonality property of the given basis to find the coordinates. Solutions 1. a) Yes, b) No. 2. E=([0,0,1],[0,1,0],[1,0,0]), 3. E=/[1 2,1 2,1 2,1 2],[0,1 ↓2,↑1 ↓2,0],[↑1 2↓3,↑1 2↓3,↑1 2↓3,↑3 2↓3], 4. x=[ 4 3,7 2,11 3]. 21 12 Determinants 1. Evaluate the determinant of the matrix A. A=  1↑21 ↑35↑7 1↑61  . 2. Evaluate the determinant of the matrix B. B=)sin x↑cos x cos xsin x*. 3. Use Cramer’s rule to find the value of the variable x2of the given system of lin. equations. x1+x2+x3+x4=↑1 2x1↑x2+3x3=1 2x2+3x4=↑1 ↑x1+2x3+x4=↑2 4. Use Cramer’s rule to find the value of the variable x3of the given system of lin. equations. 3x1↑x2+2x3=7 4x1↑3x2+2x3=4 2x1+x2+3x3=13 5. Use Cramer’s rule to find the value of the variable x1of the given system of lin. equations. 2x2+x3=2 x2+2x3+x4=1 x1+x3+2x4=4 2x1+x4=8 6. Find A→1. Use the formula with the augmented matrix ˜ A. A=  011 ↑12↑1 1↑20  7. Find B→1. Use the formula with the augmented matrix ˜ B. B=  2↑10 ↑2↑11 2↑32  22 Solutions 1. ↑16, 2. 1, 3. ↑2, 4. 3, 5. 4, 6. A→1=  22 3 11 1 0↑1↑1 , 7. B→1=↑1 4  12↑1 64↑2 84↑4 . 23 References [1] FEI, VSB-TU Ostrava, (2009), Petra Vondr´akov´a (ˇ Sarmanov´a), Sb´ırka neˇreˇsen´ych pˇr´ıklad˚u z line´arn´ı algebry, accessible from: https://homel.vsb.cz/ ber95/LA/la.htm 24