One more problem! Let L:V->R^5 be a linear transformation

[luckyc1423]

Let L: V ---> R^5 be a linear transformation.

a) If L is one-to-one and dim(Im L) = 3, what is dim(V)?
b) If dim(v) = 3 and dim(ker L) = 1, what is dim(Im L)?
c) If L is onto, what can we say about dim(V)?
d) If L is one-to-one, what can we say about the dim(V)?
e) If L is one-to-one and onto, what can we say about dim(V)?

 

[Matt]

Re: One more problem! Let L:V->R^5 be a linear transforma

Let L: V ---> R^5 be a linear transformation.

a) If L is one-to-one and dim(Im L) = 3, what is dim(V)?

If L is one-to-one, then L:V->Im(L) is an isomorphism.

b) If dim(v) = 3 and dim(ker L) = 1, what is dim(Im L)?

Look at the Dimension Theorem.

c) If L is onto, what can we say about dim(V)?
d) If L is one-to-one, what can we say about the dim(V)?
e) If L is one-to-one and onto, what can we say about dim(V)?

Look at the Dimension Theorem.