Column space, null space and rank

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Vectors are written as rows in these questions. R^n is the real vector space with n coordinates. Enter your answer as concisely as possible.

Can {(x,y): 2x+3y-6=0} be the null space of any linear map?

1 point

True

False

Clear selection

Can {(x,y): 2x^2+3y^2-6xy=0} be the range of any linear map?

1 point

True

False

Clear selection

A is a 10 x 20 matrix. Can its null space have dimension 5?

1 point

True

False

Clear selection

A = [1 2 3; 2 3 4] is a 2 x 3 matrix (semi-colon separates rows). Find a such that null(A) = span{(1,a,1)}.

2 points

A = [1 2 3; 3 4 5; 6 7 8] is a 3 x 3 matrix (semi-colon separates rows). Find dim null(A).

2 points

A = [a 2; b c] is a 2 x 2 matrix (semi-colon separates rows). Find a,b,c so that (1, -1) belongs to null(A) and (3, 15) belongs to colspace(A).

3 points

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