I believe that my answers may be some what accurate. I was hoping that some of you who are good in Linear Algebra can double check my answers. This is part of a practice test the professor gave us. I will double check with him throughout the week.
Part I. True or False. Write T if the assertion is always true, and F if the assertion is sometimes false. In true or false questions, you get +1 marks for a correct response, -1 for an incorrect respones, and 0 for no respones.
1. V = {x(vector) belongs to R(^3) :llx(vector)ll = 1 } is a subspace of R(^3).
2. R(^2) is a vector subspace of R(^3).
3. The function T : M (sub. n x n) (R) gets mapped to R given by T(A) = det A is a linear transformation.
4. The function T : M (sub. n x n) (R) gets mapped to R given by T(A) = tr(A) is a linear transformation.
5. The function T : M (sub. n x n) (R) gets mapped to M (sub. n x n) (R) given by T(A) = (tr(A))A(^Transpose) is a linear transformation.
6. The cross product of vectors in R(^3) is associative.
7. Let A be an n x n matrix. Assume that there is v(vector) that belongs to R(^n) such that the equation Ax(vector) = v(vector) doesn't have solutions x(vector) belongs to R(^n). Then the equation Ax(vector) = 0 has non-zero solutions.
8. Let V an W be finite-dimensional vector spaces and let T : V mapped to W be a linear transformation. Then T maps a basis in V to a basis in W.
9. Let V an W be finite-dimensional vector spaces and let T : V mapped to W be a linear transformation. Then T maps a linearly independent set in V to a linearly independent set in W.
10. There exists a vector space space without basis.
Put your answers in this formatt (the way I did it)(any explanations would be great) THANX.
Here are my answers:
1. Blank
2. T
3. F
4. F
5. T
6. F
7. T
8. T
9. T
10. F
Part I. True or False. Write T if the assertion is always true, and F if the assertion is sometimes false. In true or false questions, you get +1 marks for a correct response, -1 for an incorrect respones, and 0 for no respones.
1. V = {x(vector) belongs to R(^3) :llx(vector)ll = 1 } is a subspace of R(^3).
2. R(^2) is a vector subspace of R(^3).
3. The function T : M (sub. n x n) (R) gets mapped to R given by T(A) = det A is a linear transformation.
4. The function T : M (sub. n x n) (R) gets mapped to R given by T(A) = tr(A) is a linear transformation.
5. The function T : M (sub. n x n) (R) gets mapped to M (sub. n x n) (R) given by T(A) = (tr(A))A(^Transpose) is a linear transformation.
6. The cross product of vectors in R(^3) is associative.
7. Let A be an n x n matrix. Assume that there is v(vector) that belongs to R(^n) such that the equation Ax(vector) = v(vector) doesn't have solutions x(vector) belongs to R(^n). Then the equation Ax(vector) = 0 has non-zero solutions.
8. Let V an W be finite-dimensional vector spaces and let T : V mapped to W be a linear transformation. Then T maps a basis in V to a basis in W.
9. Let V an W be finite-dimensional vector spaces and let T : V mapped to W be a linear transformation. Then T maps a linearly independent set in V to a linearly independent set in W.
10. There exists a vector space space without basis.
Put your answers in this formatt (the way I did it)(any explanations would be great) THANX.
Here are my answers:
1. Blank
2. T
3. F
4. F
5. T
6. F
7. T
8. T
9. T
10. F
峠 で ドリフト を して 下さい。
俺 は 
...
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