OpenStudy (anonymous):

The subspace X of the real vector space R^3 is given as the following span of a set of vectors: X = span(1; 1; 0); (2; 5; 14); (0; 1; 2); (5; 0; 10) Determine a basis for X as well as the dimension of X. (d) Let V and W be two real vector spaces and let T : V > W be a linear transformation. Define the kernel of T and prove that it is a subspace of V .

7 years ago
OpenStudy (anonymous):

For the first part, make those vectors the columns of a matrix, and row reduce the matrix to the reduced echelon form. Count the pivots. Thats the dimension of X. the columns that contain the pivots will tell you which columns form the basis.

7 years ago
OpenStudy (anonymous):

For the second, they just want a general definition of the Null Space (or Kernel) of a Linear Transformation. In general, the Null Space, which i'll call N, is: $N = \left\{ \space x\space|\space T(x) = 0 \right\}$ Its all the vectors in the vector space such that T(x) = 0. It is a subspace because it is closed under addition and scalar multiplication. If u and v are in the Null Space, and c is a scalar, then (using the properties of a Linear Transformation) we obtain: $T(cu+v) = T(cu)+T(v) = cT(u)+T(v) = c*0+0 = 0$ Also, 0 is in the Null Space: $T(0) = 0$

7 years ago
OpenStudy (anonymous):

thanks alot!

7 years ago
OpenStudy (anonymous):

Reduced row echelon form: 1 -2 0 -5 0 1 1/7 5/7 0 0 0 0 so dim=4 basis:{(1,0),(-2,1),(0,1/7),(-5,5/7)}

7 years ago
OpenStudy (anonymous):

dont know how to type a matrix!

7 years ago
The text passage contains two distinct mathematical questions, one related to finding a basis and dimension of a subspace in a vector space, and the other about defining and proving the kernel of a linear transformation. Answers to both questions are provided in the subsequent posts.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
OpenStudy (anonymous):

The subspace X of the real vector space R^3 is given as the following span of a set of vectors: X = span(1; 1; 0); (2; 5; 14); (0; 1; 2); (5; 0; 10) Determine a basis for X as well as the dimension of X. (d) Let V and W be two real vector spaces and let T : V > W be a linear transformation. Define the kernel of T and prove that it is a subspace of V .

7 years ago
OpenStudy (anonymous):

For the first part, make those vectors the columns of a matrix, and row reduce the matrix to the reduced echelon form. Count the pivots. Thats the dimension of X. the columns that contain the pivots will tell you which columns form the basis.

7 years ago
OpenStudy (anonymous):

For the second, they just want a general definition of the Null Space (or Kernel) of a Linear Transformation. In general, the Null Space, which i'll call N, is: $N = \left\{ \space x\space|\space T(x) = 0 \right\}$ Its all the vectors in the vector space such that T(x) = 0. It is a subspace because it is closed under addition and scalar multiplication. If u and v are in the Null Space, and c is a scalar, then (using the properties of a Linear Transformation) we obtain: $T(cu+v) = T(cu)+T(v) = cT(u)+T(v) = c*0+0 = 0$ Also, 0 is in the Null Space: $T(0) = 0$

7 years ago
OpenStudy (anonymous):

thanks alot!

7 years ago
OpenStudy (anonymous):

Reduced row echelon form: 1 -2 0 -5 0 1 1/7 5/7 0 0 0 0 so dim=4 basis:{(1,0),(-2,1),(0,1/7),(-5,5/7)}

7 years ago
OpenStudy (anonymous):

dont know how to type a matrix!

7 years ago
user
The text passage contains two distinct mathematical questions, one related to finding a basis and dimension of a subspace in a vector space, and the other about defining and proving the kernel of a linear transformation. Answers to both questions are provided in the subsequent posts.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS