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_forwardLet A be an n x n matrix, let S = (ü, ü, u) be a set of non-zero vectors in R", and let U be a subspace of R" of dimension at least 1. Look at the expressions and phrases that follow. Select the ones that DO NOT make sense, because they either equate two different "types" of thing that can't be equal, use nonsensical notation, or try to perform an operation that is not defined. (For the computer programmers reading this, the question is essentially "find the type errors".) Warning: To clarify, you're not being asked which ones are true. You're being asked to identify which equations don't make sense. From the 11 choices, select all that apply "a basis of 5" "the solutions of the system of equations" im(A) (R" Ay for some FR") "the span of A" im(A) - (A#-5) "S spans U "the solutions of the matrix" im(A) {ER: Af=ÿ) null(A) = {A=0} 1 null(4)-(ER": Až=6) "U spans S"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
- If 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_forwardFor 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_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
- -5 -4 , find the closest point to v in the subspace W spanned by -2 -2 and Given v = 7 -3 70arrow_forwardConsider the subspaces U = span{[401],[4 1 -4] 3 3], [-54 -2]} W = span{ [-5 3arrow_forwardDetermine 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_forward
- If 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_forward1. 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_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning