1a) Asks you to find "the image by f of the vector
". You do that by the matrix multiplication
1b) Asks you to find the
determinant of the matrix: use row operations to find
.
1c) A matrix is "invertible" if and only if its determinant is
not 0.
For 2a, I would use "indirect proof"- assume that the result is
not true. That is, assume there exist a, b, and c, not all 0, such that
. That is, such that
. That is, that
is in rhe kernel of f. Now if the kernel consists
only of the 0 vector, we must have
. Since you are given that
,
, and
are independent, what does that tell you about a, b, and c?
A basis, S, for an n dimensional vector space, V, has 3 properties:
1) the vectors in S are linearly independent.
2) the vectors in S span V.
3) there are n vectors in S.
And any
two of those imply the third! So if you know (from 2a) that these vectors are independent and you know that there are 3 vector while the dimension of the space is 3, then it follows that they also span V and form a basis.