True or False. Circle your answer. T F: The plane x + y + z = 0 in R^3 is a subs

Question

True or False. Circle your answer. T F: The plane x + y + z = 0 in R^3 is a subspace of R^3. T F: The plane x + y + z = 10 in R^3 is a subspace of R^3. T F: rank (A) = number of columns - number of free variables. T F: For any m times n matrix A and any real number k, rank (kA) = A; (rank(A)). T F: The columns of a 5 times 7 matrix A are always linearly dependent. T F: For any m times n matrix A; C(A) = C(A^T). T F: If B = {u, v, w} is basis for a 3-dimensional vector space V, then B' = {au, bv, cw} is also a basis for V, where a, b, c are any non-zero scalars. T F: Let A be a 5 times 9 matrix such that dim(C(A)) = 4, then dim(N(A)) = 1. T F: For any 3 times 3 matrix A; C (A) notequalto N (A). T F: Any subset of R^7 consisting of 8 vectors is linearly dependent. Answer the followings (you do not need to show your work). Write 1 times 3 matrix A whose null space is the plane 2x - 3y + z = 0 Write down a matrix A such that N (A) is the set of all linear combinations of (2, 2, 1, 0) and (3, 1, 0, 1) Construct a matrix A whose column space contains (1, 2, 1) and (1, 0, 1) and the null space contains (3, 1, 1). Write two linearly independent vectors u and v on the plane x - 2y + 3z = 0. Write a basis for the vector space consisting of all 2 times 2 symmetric matrices.

Explanation / Answer

a) x+y +z is a subspace as it follows the additiona and scalr multiplication property and 0 is a

vector in the subspace

True

b) x+y +z = 10

Its is not a subspace as there is np zero vector present in sthe subspace

c) false ; rank = no. of rows - no. of free variables

d) False Rank(kA) = rank (A)

e) True :

B) a) 2x -3y +z =0

x = 3y/2 - z/2 ---> [ 2 -3 z ] ( 3 x1 matrix)

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