For Problems 25-31, determine a linearly independent set of
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- Find a vector that spans the kernel of the following matrix: [ 1 0 2 4 ][ 0 1 -3 -1 ][ 3 4 -6 8 ][ 0 -1 3 4 ]arrow_forwardIf A is m by n, how many separate multiplications are involved when(a) A multiplies a vector x with n components?(b) A multiplies an n by p matrix B?( c) A multiplies itself to produce A2 ? Here m = n.arrow_forwardGiven V = -5 2 2 6 3 6 6 -1 8-9 and -6 -4 36 find the closest point to in the subspace W spanned byarrow_forward
- For each of the following lists of vectors in R3, determine whether the first vector can be expressed as a linear combination of the other two. (a) (-2,0,3) ,(1,3,0),(2,4,-1) (b) (1,2,-3) ,(-3,2,1) ,(2,-1,-1) (c) (3,4,1) ,(1,-2,1), (-2,-1,1) (d) (2,-1,0) , (1,2,-3), (1,-3,2) (e) (5,1,-5) , (1,-2,-3), (-2,3,-4) (f) (-2,2,2) ,(1,2,-1) ,(-3,-3,3)arrow_forwardIf k is a real number, then the vectors (1,k),(k,7k+8) are linearly independent precisely when k ≠ a,b, where a = ? and b = ?, and a<barrow_forward-5 -4 , find the closest point to v in the subspace W spanned by -2 -2 and Given v = 7 -3 70arrow_forward
- Determine the dimensions of the following subspaces of R4 All vectors of the form (a, b, c, d), where d = a + b y c = a – b. All vectors of the form (a, b, c, d), where a = b = c = d. Note: In the image the problem is described more clearly, do not skip any step and solve the two parts a and b.arrow_forwardConsider the subspaces U = span{[401],[4 1 -4] 3 3], [-54 -2]} W = span{ [-5 3arrow_forwardThe integers 1 through 36 (inclusive) are used once each as the coefficients for six vectors in P 5 . What is the smallest dimensional subspace that can be spanned by the six vectors? Wholly justify your answer.arrow_forward
- 1. a. b. x1 + x2 x1 2x1 T X2 Find the solution set of the following system of linear equations : 5x3 2x3 X3 || 3 1 0 What is the dimension of this set of solutions? Of what vector space is it a linear subspace? (R"?)arrow_forwardFind the angle between the vectors [ 1 ] And [ -3 ][ 3 ] [ 2 ][ 2 ] [ 5 ]arrow_forwardIf possible, find a linear combination of the form w = a₁v₁ + a₂₂ + 3⁄³ where v₁ = (2, −1, 4), v₂ = (3, 0, 1), v3 = (1, 2, −1), and w = (-7, 1, 5). (Give a, a, and a3 as real numbers. If w cannot be written as a linear combination of the other three vectors, enter DNE.) (₁₁²₂₁²3) =arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning