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lrm;Table of Contents
- 1. new schedule today :/
- 2. Systems of equations, matrix equations, and vectors
- 3. in class work! See ./KBe20math530srcNull_space_and_column_space_intro.pdf
- 3.1. \(A=\begin{pmatrix}1&0\\0&1\end{pmatrix}\)
- 3.2. \(A=\begin{pmatrix}1&0&0\\0&1&0\end{pmatrix}\)
- 3.3. \(A = \begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}\)
- 3.4. \(A = \begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix}\)
- 3.5. \(A = \begin{pmatrix}1&0&0\\0&1&0\\0&1&0\end{pmatrix}\)
- 3.6. \(A = \begin{pmatrix}0&0&0\\0&0&3\\0&0&0\end{pmatrix}\)
- 3.7. \(A = \begin{pmatrix}1&2&-1\\1&-1&0\\3&3&-2\end{pmatrix}\)
- 4. Then we talked about some stuff
1 new schedule today :/
2 Systems of equations, matrix equations, and vectors
3 in class work! See ./KBe20math530srcNull_space_and_column_space_intro.pdf
3.1 \(A=\begin{pmatrix}1&0\\0&1\end{pmatrix}\)
3.1.1 How many solutions \(x\) satisfy \(Ax=0\)?
The only solution is x=0, because \(Ax = x\).
3.1.2 When the answer is "infinitely many" what tools might we have to describe the size of that set?
N/A
3.1.3 How many possible outcomes \(b\) are there for the equation \(Ax=b\) for any \(x\).
There can be infintely many vaules of \(b\)..? The vector space is dim 2
3.2 \(A=\begin{pmatrix}1&0&0\\0&1&0\end{pmatrix}\)
3.2.1 How many solutions \(x\) satisfy \(Ax=0\)?
Infinitely many (anything of the form \(\begin{pmatrix}0\\0\\x\end{pmatrix}\))
3.2.2 When the answer is "infinitely many" what tools might we have to describe the size of that set?
A column in the matrix is zero? Maybe the columns are linearly dependent. Input is dim 1
3.2.3 How many possible outcomes \(b\) are there for the equation \(Ax=b\) for any \(x\).
Infinite with \(\text{dim} 2\)?
3.3 \(A = \begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}\)
3.3.1 How many solutions \(x\) satisfy \(Ax=0\)?
Only one value of \(x\) makes the product zero.
3.3.2 When the answer is "infinitely many" what tools might we have to describe the size of that set?
n/a
3.3.3 How many possible outcomes \(b\) are there for the equation \(Ax=b\) for any \(x\).
column vector has dimension 3, but the vector space has dim 2
3.4 \(A = \begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix}\)
3.4.1 How many solutions \(x\) satisfy \(Ax=0\)?
infinite, same vectors as subproblem 2
3.4.2 When the answer is "infinitely many" what tools might we have to describe the size of that set?
dimension 2? column vectors in the matrix are linearly dependent.
3.4.3 How many possible outcomes \(b\) are there for the equation \(Ax=b\) for any \(x\).
infinite, dim 2 (but each vector is dim 3)
3.5 \(A = \begin{pmatrix}1&0&0\\0&1&0\\0&1&0\end{pmatrix}\)
3.5.1 How many solutions \(x\) satisfy \(Ax=0\)?
infinite, vectors of the form \(\begin{pmatrix}0\\a\\-a\end{pmatrix}\) (columns linearly dependent)
3.5.2 When the answer is "infinitely many" what tools might we have to describe the size of that set?
dimension 2 subspace of \(\mathbb F^3\)
3.5.3 How many possible outcomes \(b\) are there for the equation \(Ax=b\) for any \(x\).
infinite, dim2 subspace of \(\mathbb F^3\)
3.6 \(A = \begin{pmatrix}0&0&0\\0&0&3\\0&0&0\end{pmatrix}\)
3.6.1 How many solutions \(x\) satisfy \(Ax=0\)?
infinite, vectors of the form \(\begin{pmatrix}a\\b\\0\end{pmatrix}\) (columns linearly dependent)
3.6.2 When the answer is "infinitely many" what tools might we have to describe the size of that set?
dim 2 subspace of \(\mathbb F^3\)
3.6.3 How many possible outcomes \(b\) are there for the equation \(Ax=b\) for any \(x\).
output has dim 1
3.7 \(A = \begin{pmatrix}1&2&-1\\1&-1&0\\3&3&-2\end{pmatrix}\)
3.7.1 How many solutions \(x\) satisfy \(Ax=0\)?
Seems like the rows are linearly independent, so it should be just 1 solution \(x=0\)? infinite, vectors of the form \(\begin{pmatrix}a\\b\\0\end{pmatrix}\) (columns linearly dependent)
3.7.2 When the answer is "infinitely many" what tools might we have to describe the size of that set?
dim 0
3.7.3 How many possible outcomes \(b\) are there for the equation \(Ax=b\) for any \(x\).
output should be dim 3
4 Then we talked about some stuff
4.2 The null space is the stuff that gets sent to zero (responses to subpart 1) definition
See Null Spaces