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]
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related searches
- basic math practice test questions
- sat math practice test printable
- 3rd grade math state test practice
- 2nd grade math assessment test printable
- college math placement test practice
- free math placement test practice
- math placement test practice problems
- free math assessment test online
- ftce math practice test free
- math state test 2019
- math analytics test for employers
- sat math practice test pdf