T/F

(AB)T =AT ∗BT

FALSE (AB)T =BT ∗AT

A mapping T: Rn →Rm is onto Rm if every vector x in Rn maps onto some vector in Rm.

FALSE A linear transformation is onto if the codomain is equal to the range

Not every linear transformation from Rn to Rm is a matrix transformation.

FALSE For a linear transformation from Rn to Rm we see where the basis vector in Rn get mapped to. These form the standard matrix.

If A is a 3x2 matrix, then the transformation x→Ax cannot be one-to-one.

FALSE Since the transformation maps from R2 to R3 and 2 < 3, it can be one-to-one but not onto.

Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B.

FALSE Swap A and B then its true

The transpose of a product of matrices equals the product of their tranposes in the same order.

FALSE The transpose of a product of matrices equals the product of their tranposes in the reverse order.

When two linear transformations are performed one after another, then combined effect may not always be a linear transformation.

FALSE check properties of a linear transformation.

The columns of the standard matrix for a linear transformation from Rn to Rm are the images of the columns of the nxn identity matrix.

TRUE

The transpose of a sum of matrices equals the sum of their transposes.

TRUE

AB+AC=A(B+C)

TRUE Matrix multiplication distributes over addition.

AT + BT = (A + B)T

TRUE See properties of transposition. Also should be able to think through to show this. When we add we add corresponding entries, these will remain corresponding entries after transposition.

A linear transformation T : Rn → Rm is completely determined by its effect on columns of the n × n identity matrix

TRUE The columns on the identity matrix are the basis vectors in Rn. Since every vector can be written as a linear combination of these, and T is a linear transformation, if we know where these columns go, we know everything.

If A is a 3×2 matrix, then the transformation x→Ax cannot map R2 onto R3

TRUE You can not map a space of lower dimension onto a space of higher dimension

If T : R2 → R2 rotates vectors about the origin through an angle φ, then T is a linear transformation.

TRUE show that the properties of linear transformations are preserved under rotations.

Null space vs column space?

The dimension of the null space is the number of non-pivot columns of A and the dimension of the column space is the number of pivot columns of A.