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.