Could a set of three vectors in R* span all of R*? Explain. What about n vectors in R" when n is less than m? Could a set of three vectors in R* span all of R*? Explain. Choose the correct answer below. O A. Yes. A set of n vectors in R" can span R" when n
Could a set of three vectors in R* span all of R*? Explain. What about n vectors in R" when n is less than m? Could a set of three vectors in R* span all of R*? Explain. Choose the correct answer below. O A. Yes. A set of n vectors in R" can span R" when n
Could a set of three vectors in R* span all of R*? Explain. What about n vectors in R" when n is less than m? Could a set of three vectors in R* span all of R*? Explain. Choose the correct answer below. O A. Yes. A set of n vectors in R" can span R" when n
Transcribed Image Text:**Question:**
Could a set of three vectors in \(\mathbb{R}^4\) span all of \(\mathbb{R}^4\)? Explain. What about \(n\) vectors in \(\mathbb{R}^m\) when \(n\) is less than \(m\)?
Could a set of three vectors in \(\mathbb{R}^4\) span all of \(\mathbb{R}^4\)? Explain. Choose the correct answer below.
**Options:**
- **A.** Yes. A set of \(n\) vectors in \(\mathbb{R}^m\) can span \(\mathbb{R}^m\) when \(n < m\). There is a sufficient number of rows in the matrix \(A\) formed by the vectors to have enough pivot points to show that the vectors span \(\mathbb{R}^m\).
- **B.** Yes. Any number of vectors in \(\mathbb{R}^4\) will span all of \(\mathbb{R}^4\).
- **C.** No. There is no way for any number of vectors in \(\mathbb{R}^4\) to span all of \(\mathbb{R}^4\).
- **D.** No. The matrix \(A\) whose columns are the three vectors has four rows. To have a pivot in each row, \(A\) would have to have at least four columns (one for each pivot).
Transcribed Image Text:Suppose \( A \) is a \( 4 \times 3 \) matrix and \( b \) is a vector in \( \mathbb{R}^4 \) with the property that \( Ax = b \) has a unique solution. What can you say about the reduced echelon form of \( A \)? Justify your answer.
Choose the correct answer below.
- **A.** The first term of the first row will be a 1 and all other terms will be 0. There is only one variable \( x_m \), so there is only one possible solution.
- **B.** There will be a pivot position in each row. If a row did not have a pivot position then the equation \( Ax = b \) would be inconsistent.
- **C.** The first 3 rows will have a pivot position and the last row will be all zeros. If a row had more than one 1, then there would be an infinite number of solutions for \( a_m x_m = b_m \).
- **D.** The first row will have a pivot position and all other rows will be all zeros. There is only one equation to solve, so there is only one solution.
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
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