2. State \"True\" or \"False\" for each of the following statements. You do not

Question

2. State "True" or "False" for each of the following statements. You do not need to justify your answers. (f) Any vector space of dimension 2 has infinitely many subspaces. (g) Any vector space of dimension 3 can be expanded by four vec- (h) Any vector space of dimension 3 can be expanded by two vec- (i) Three vectors are linearly dependent if and only if one of them The column space and row space of the same matrix A will tors tors can be written as a linear combination of the other two. have the same dimension

Explanation / Answer

f) true

g) true

h) true

i) true

(Suppose {x1, x2, x3} are linearly dependent. Then there exist scalar constants c1, c2, c3 such that c1x1 + c2x2 + c3x3 = 0, where at least one of the constants is not zero

If, say, c2 is non zero , then we can solve for x2: x2 = (1/c2)(c1x1 + c3x3). )

j.) true

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