(完整版)哈工大选修课LINEARALGEBRA试卷及答案,推荐文档

LINEAR ALGEBRA

AND

ITS APPLICATIONS 姓名：易

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成绩：

1. Definitions

(1) Pivot position in a matrix;

(2) Echelon Form;

(3) Elementary operations;

(4) Onto mapping and one-to-one mapping;

(5) Linearly independence.

2. Describe the row reduction algorithm which produces a matrix in reduced echelon form.

3. Find the matrix that corresponds to the composite transformation of a scaling by 0.3, 33⨯a rotation of , and finally a translation that adds (-0.5, 2) to each point of a figure.

90︒4. Find a basis for the null space of the matrix

361171223124584A ---⎡⎤⎢⎥=--⎢⎥⎢⎥--⎣⎦

5. Find a basis for Col of the matrix

A 1332-9-2-22-822307134-111-8A ⎡⎤⎢⎥⎢⎥=⎢⎥⎢⎥⎣⎦

6. Let and be positive numbers. Find the area of the region bounded by the ellipse a b whose equation is

22

22

1x y a b +=7. Provide twenty statements for the invertible matrix theorem.

8. Show and prove the Gram-Schmidt process.

9. Show and prove the diagonalization theorem.

10. Prove that the eigenvectors corresponding to distinct eigenvalues are linearly independent.

Answers:

1. Definitions

(1) Pivot position in a matrix:

A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is a column of A that contains a pivot position.

(2) Echelon Form:

A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties:

1. All nonzero rows are above any rows of all zeros.

2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.

3. All entries in a column below a leading entry are zeros.

If a matrix in a echelon form satisfies the following additional conditions, then it is in reduced echelon form (or reduced row echelon form):

4. The leading entry in each nonzero row is 1.

5. Each leading 1 is the only nonzero entry in its column.

(3) Elementary operations:

Elementary operations can refer to elementary row operations or elementary column operations.

There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations):

1. (Replacement) Replace one row by the sum of itself anda multiple of another row.

2. (Interchange) Interchange two rows.

3. (scaling) Multiply all entries in a row by a nonzero constant.

(4) Onto mapping and one-to-one mapping:

A mapping T : R n → R m is said to be onto R m if each b in R m is the image of at least one x in R n.

A mapping T : R n → R m is said to be one-to-one if each b in R m is the image of at most one x in R n.

(5) Linearly independence:

An indexed set of vectors {V1, . . . ,V p} in R n is said to be linearly independent if the vector equation