4.7 – Planes in R3

4.8 – Span – Use RREF

4.9 – Subspaces, basses, dimension – Use RREF

4.10 Row, Column, Null Spaces – Use RREF again

5.1 Linear Transformations

True or False:
1. A set of vectors can still be a valid span without being a basis of the subspace.

2. The number of columns in a matrix is the sum of the dimension of the column or row space and the dimension of the null space.

3. If rref(A) = rref(B) then col(A) = col(B)

4. If rref(A) = rref(B) then row(A) = row(B)

5. A linear transform such that T(x)=Ax can have A=(2x,2,5z)

6. A linear transform such that T(x)=Ax can have A=(2x,2y+2,5z)

7. A linear transform such that T(x)=Ax can have A=(|2x|,4y,5z)

8. A linear transform such that T(x)=Ax can have A=(x^2,4y,5z)

Give an example of:
1. A set of vectors that are both a span and basis for R3

2. A vector perpendicular to the XY plane

3. A vector normal to the Z axis

4. A subset of vectors in R3 that do not make up a valid subspace of R3

5. A square matrix that does not span R3

6. A subset of vectors that form a basis for R3 given equation -x-2y+3z=2.