Problem 2. Problem 3. Perform, if possible, each matrix-vector multiplication. Answer. 3. Not defined. Solve this matrix equation.

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1 Problem 2 Perform, if possible, each matrix-vector multiplication Not defined. Problem 3 Solve this matrix equation.

2 Matrix-vector multiplication gives rise to a linear system. Gaussian reduction shows that:,, and. Problem 5 Assume that is determined by this action. Using the standard bases, find 1. the matrix representing this map; 2. a general formula for. Again, as recalled in the subsection, with respect to (canonical or standard basis), a column vector represents itself. 1. To represent with respect to we take the images of the basis vectors from the domain, and represent them with respect to the basis for the codomain.

3 These are adjoined to make the matrix. 2. For any in the domain, and so is the desired representation. Problem 11 Express this transformation with respect to the standard basis (canonical): the dilation map, which multiplies all vectors by the same scalar The picture of is this.

4 This map's effect on the vectors in the standard basis for the domain is and those images are represented with respect to the codomain's basis (again, the standard basis) by themselves. Thus the representation of the dilation map is this. Problem 16 Find the Image (or range) of the linear transformation of represented with respect to the standard bases by each matrix. 1.

5 2. 3. a matrix of the form 1. It is the set of vectors of the codomain represented with respect to the codomain's basis in this way. As the codomain's basis is, and so each vector is represented by itself, the IMAGE (or range) of this transformation is the -axis. 2. It is the set of vectors of the codomain represented in this way. With respect to vectors represent themselves so this image is the axis. 3. The set of vectors represented with respect to as Problem 19 is the line, provided either or is not zero, and is the set consisting of just the origin if both are zero. 1. Rotation of all vectors in three-space through an angle about the -axis is a transformation of. Represent it with respect to the standard bases. Arrange the rotation so that to someone whose feet are at the origin and whose head is at, the movement appears clockwise. 2. Repeat the prior item, only rotate about the -axis instead. (Put the person's head at.)

6 3. Repeat, about the -axis. 1. The picture is this. The images of the vectors from the domain's basis are represented with respect to the codomain's basis (again, ) by themselves, so adjoining the representations to make the matrix gives this. 2. The picture is similar to the one in the prior answer. The images of the vectors from the domain's basis

7 are represented with respect to the codomain's basis by themselves, so this is the matrix. 3. To a person standing up, with the vertical -axis, a rotation of the -plane that is clockwise proceeds from the positive -axis to the positive The images of the vectors from the domain's basis -axis.. are represented with respect to by themselves, so the matrix is this: Problem 1 Decide if the vector is in the column space of the matrix. 1., 2.,

8 3., 1. Yes; we are asking if there are scalars and such that which gives rise to a linear system and Gauss' method produces the matrix. and. That is, there is indeed such a pair of scalars and so the vector is indeed in the column space of 2. No; we are asking if there are scalars and such that and one way to proceed is to consider the resulting linear system that is easily seen to have no solution. Another way to proceed is to note that any linear combination of the columns on the left has a second component half as big as its first component, but the vector on the right does not meet that criterion. 3. Yes; we can simply observe that the vector is the first column minus the second. Or, failing that, setting up the relationship among the columns

9 and considering the resulting linear system gives the additional information (beyond thatt there is at least one solution) that there are infinitely many solutions. Parametizing gives and, and so taking to be zero gives a particular solution of,, and (which is, of course, the observation made at the start). Problem 11 Describe geometrically the action on of the map represented with respect to the standard bases by this matrix: Do the same for these. The first map

10 stretches vectors by a factor of three in the direction and by a factor of two in the direction. The second map projects vectors onto the axis. The third interchanges first and second components (that is, it is a reflection about the line ). The last stretches vectors parallel to the axis, by an amount equal to three times their distance from that axis.