Name:__________________ Math 2318 Test 2



Math 2318 Review 4(answers)

1. Use the Gram-Schmidt orthonormalization process to transform [pic] into an orthonormal basis for [pic].

[pic], [pic],

[pic]

[pic], [pic]

[pic]

[pic]

2. Find the orthogonal complement [pic] of the subspace S of [pic] spanned by the two column vectors of [pic].

We need to find the null space of [pic]: [pic]

[pic], so [pic].

3. Find the projection of the vector [pic] onto the subspace [pic].

It’s easy to convert the basis for S into an orthonormal basis for S. [pic], so [pic]

The projection could also be found using the least squares solution of [pic].

4. Find bases for the four fundamental subspaces of the matrix [pic].

[pic]: [pic], [pic], so a basis for [pic] is [pic].

[pic] is the span of the columns of [pic], which is the span of the rows of A, and row reducing A leads to [pic], so a basis for [pic] is [pic].

[pic] is the span of the columns of A, so row reduce [pic]. [pic], so a basis for [pic] is [pic].

[pic]: [pic], [pic], so a basis for [pic] is [pic].

5. Find the least squares regression line for the set of points [pic].

The system to solve is [pic]. As a matrix equation it’s [pic]. Multiplying both sides by the transpose of the coefficient matrix leads to [pic] which simplifies into [pic]. The solution is [pic], and the least squares line is [pic].

6. Find the least squares linear approximation in [pic] to [pic].

An orthonormal basis for the linear functions in [pic] is [pic], so the least squares linear approximation is given by [pic].

7. Find the least squares quadratic approximation in [pic] to [pic].

An orthonormal basis for the quadratic functions in [pic] is [pic], so the least squares quadratic approximation is given by [pic]

8. Determine if the given function is a linear transformation:

a) [pic], [pic]

[pic], since [pic].

b) [pic], [pic]

[pic], since [pic]

and

[pic]

9. Consider the linear transformation [pic], [pic].

a) Find the standard matrix representation of T.

[pic]

b) Find a basis for the kernel of T.

[pic], [pic], so a basis for the kernel of T is [pic].

c) Find a basis for the range of T.

We can row reduce [pic]: [pic], so a basis for the range of T is [pic].

d) Find the rank of T.

The rank of T is [pic].

e) Find the nullity of T.

The nullity of T is [pic].

10. Given [pic] is a linear transformation, and [pic].

a) Find the [pic].

[pic], so [pic], and hence [pic].

b) Is T an onto function? Explain.

[pic], the image of T is a three dimensional subspace of [pic], but the only three dimensional subspace of [pic] is [pic]. So T is onto.

11. Consider the linear transformation [pic],[pic].

a) Find the standard matrix representation, A, of T.

[pic]

b) Find the matrix representation, C, relative to [pic]and [pic] for the bases [pic] and [pic].

[pic], [pic], so [pic]

c) Find the transition matrix, S, from the basis B to the standard basis [pic].

[pic], so [pic]

d) Find the transition matrix, T, from the basis [pic] to the basis [pic].

[pic], so [pic]

e) Show that [pic].

[pic]

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