Without any computation, determine whether the following statements are true. S

Question

Without any computation, determine whether the following statements are true. S = {[1 0 2 -1], [0 1 1 1], [2 1 1 1], [1 -1 1 1], [0 0 0 1]} in M _2 times 2 is a linearly independent set. B = {1 + t - t^2, t + 2t^2) spans P _2, the space of polynomials of degree 2 or less. The rank of a 5 times 8 matrix can be 8. The rank of an n times n invertible matrix is equal to n. There are always 5 linearly independent vectors in the row space of any 5 times 5 matrix. If the linear system Ax = b is consistent, then b must be in the column space of A

Explanation / Answer

a)

No.

M2x2 has dimension 4 and this set has 5 vectors so the set cannot be linearly independent

b)

No.

P2 has dimension 3 so this set with two vectors cannot span P2

c)

No.

Rank of a matrix=dim column space=dim row space

Since there are only 5 rows so rank cannot be larger than 5

d)

True

e)

False.

Rank can vary from 0 to 5

f)

Yes

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