text: 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
explanation: The document contains a math question and its corresponding answer. The question is about determining a basis for a subspace \( X \) of \( \mathbb{R}^3 \) and the dimension of \( X \), as well as defining the kernel of a linear transformation and proving it is a subspace. The answer provides a detailed step-by-step solution for both parts of the question.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 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
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its corresponding answer. The question is about determining a basis for a subspace \( X \) of \( \mathbb{R}^3 \) and the dimension of \( X \), as well as defining the kernel of a linear transformation and proving it is a subspace. The answer provides a detailed step-by-step solution for both parts of the question.
  
  Label: question-and-answer