(完整版)哈工大选修课LINEARALGEBRA试卷及答案,推荐文档
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(5) Linearly independence: An indexed set of vectors {V1, . . . ,Vp} in Rn is said to be linearly independent if the vector equation
(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.
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.
(4) Onto mapping and one-to-one mapping: A mapping T : Rn → Rm is said to be onto Rm if each b in Rm is the image of at least one x in Rn. A mapping T : Rn → Rm is said to be one-to-one if each b in Rm is the image of at most one x in Rn.
LINEAR ALGEBRA AND
ITS APPLICATIONS
姓名：易 学号： 成绩：
1. Definitions
(1) Pivot position in a matrix;
(2) Echelon Form;
(3) Elementary operations;
(4) Onto mapping and one-to-one mapping;
源自文库
a rotation of 90 , and finally a translation that adds (-0.5, 2) to each point of a figure.
4. Find a basis for the null space of the matrix
A
3 1
6 2
1 1 23
(5) Linearly independence.
2. Describe the row reduction algorithm which produces a matrix in reduced echelonform.
3.Find the 3 3 matrix that corresponds to the composite transformation of a scaling by 0.3,
71
2 4 5 8 4
5. Find a basis for Col A of thematrix
1 3 3 2 -9 -2 -2 2 -8 2
A
2
3
0
7
1
3
4 -1 11 -8
6. Let a and b be positive numbers. Find the area of the region bounded by the ellipse
(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.
whose equation is
x2 y2 a2 b2 1
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.
(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.
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.
(4) Onto mapping and one-to-one mapping: A mapping T : Rn → Rm is said to be onto Rm if each b in Rm is the image of at least one x in Rn. A mapping T : Rn → Rm is said to be one-to-one if each b in Rm is the image of at most one x in Rn.
LINEAR ALGEBRA AND
ITS APPLICATIONS
姓名：易 学号： 成绩：
1. Definitions
(1) Pivot position in a matrix;
(2) Echelon Form;
(3) Elementary operations;
(4) Onto mapping and one-to-one mapping;
源自文库
a rotation of 90 , and finally a translation that adds (-0.5, 2) to each point of a figure.
4. Find a basis for the null space of the matrix
A
3 1
6 2
1 1 23
(5) Linearly independence.
2. Describe the row reduction algorithm which produces a matrix in reduced echelonform.
3.Find the 3 3 matrix that corresponds to the composite transformation of a scaling by 0.3,
71
2 4 5 8 4
5. Find a basis for Col A of thematrix
1 3 3 2 -9 -2 -2 2 -8 2
A
2
3
0
7
1
3
4 -1 11 -8
6. Let a and b be positive numbers. Find the area of the region bounded by the ellipse
(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.
whose equation is
x2 y2 a2 b2 1
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.