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4.5: Geometric Meaning of Scalar Multiplication

https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/04%3A_R/4.05%3A_Geometric_Meaning_of_Scalar_Multiplication

Then, by using Definition 4.4.1, the length of this vector is given by \[\sqrt{\left( \left( k u_{1}\right) ^{2}+\left( k u_{2}\right) ^{2}+\left( k u_{3}\right) ^{2}\right) }=\left\vert k \right\vert...Then, by using Definition 4.4.1, the length of this vector is given by

$\sqrt{\left( \left( k u_{1}\right) ^{2}+\left( k u_{2}\right) ^{2}+\left( k u_{3}\right) ^{2}\right) }=\left\vert k \right\vert \sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}\nonumber$ Thus the following holds.

$\| k \vec{u} \| =\left\vert k \right\vert \| \vec{u} \|\nonumber$ In other words, multiplication by a scalar magnifies or shrinks the length of the vector by a factor of

$\left\vert k \right\vert$ .

9.4: Well Ordering and Induction

https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/09%3A_Appendices/9.04%3A_Well_Ordering_and_Induction

Let

$T$ consist of all integers larger than or equal to

$a$ which are not in

$S.$ The theorem will be proved if

$T=\emptyset .$ If

$T\neq \emptyset$ then by the well ordering principle, ther...Let

$T$ consist of all integers larger than or equal to

$a$ which are not in

$S.$ The theorem will be proved if

$T=\emptyset .$ If

$T\neq \emptyset$ then by the well ordering principle, there would have to exist a smallest element of

$T,$ denoted as

$b.$ It must be the case that

$b>a$ since by definition,

$a\notin T.$ Thus

$b\geq a+1$ , and so

$b-1\geq a$ and

$b-1\notin S$ because if

$b-1\in$

$S,$ then

$b-1+1=b\in S$ by the assumed property of

$S.$ Therefore, \(b-…

1.4: Uniqueness of the Reduced Row-Echelon Form

https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/01%3A_Systems_of_Equations/1.04%3A_Uniqueness_of_the_Reduced_Row-Echelon_Form

As we have seen in earlier sections, we know that every matrix can be brought into reduced row-echelon form by a sequence of elementary row operations. Here we will prove that the resulting matrix is ...As we have seen in earlier sections, we know that every matrix can be brought into reduced row-echelon form by a sequence of elementary row operations. Here we will prove that the resulting matrix is unique; in other words, the resulting matrix in reduced row-echelon does not depend upon the particular sequence of elementary row operations or the order in which they were performed.

2.10: LU Factorization

https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/02%3A_Matrices/2.10%3A_LU_Factorization

An LU factorization of a matrix involves writing the given matrix as the product of a lower triangular matrix (L) which has the main diagonal consisting entirely of ones, and an upper triangular matri...An LU factorization of a matrix involves writing the given matrix as the product of a lower triangular matrix (L) which has the main diagonal consisting entirely of ones, and an upper triangular matrix (U) in the indicated order.

2.8: Elementary Matrices

https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/02%3A_Matrices/2.08%3A_Elementary_Matrices

We now turn our attention to a special type of matrix called an elementary matrix.

5.8: The Kernel and Image of a Linear Map

https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/05%3A_Vector_Spaces/5.08%3A_The_Kernel_and_Image_of_a_Linear_Map

Here we consider the case where the linear map is not necessarily an isomorphism. First here is a definition of what is meant by the image and kernel of a linear transformation.

4.9: The Cross Product

https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/04%3A_R/4.09%3A_The_Cross_Product

Recall that the dot product is one of two important products for vectors. The second type of product for vectors is called the cross product.

5.9: The Matrix of a Linear Transformation

https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/05%3A_Vector_Spaces/5.09%3A_The_Matrix_of_a_Linear_Transformation

You may recall from Rn that the matrix of a linear transformation depends on the bases chosen. This concept is explored in this section, where the linear transformation now maps from one arbitrary v...You may recall from Rn that the matrix of a linear transformation depends on the bases chosen. This concept is explored in this section, where the linear transformation now maps from one arbitrary vector space to another.

1.3: Gaussian Elimination

https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/01%3A_Systems_of_Equations/1.03%3A_Gaussian_Elimination

The work we did in the previous section will always find the solution to the system. In this section, we will explore a less cumbersome way to find the solutions. First, we will represent a linear sys...The work we did in the previous section will always find the solution to the system. In this section, we will explore a less cumbersome way to find the solutions. First, we will represent a linear system with an augmented matrix. A matrix is simply a rectangular array of numbers. The size or dimension of a matrix is defined as m×n where m is the number of rows and n is the number of columns.

6.3: Properties of Linear Transformations

https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/06%3A_Linear_Transformations/6.03%3A_Properties_of_Linear_Transformations

Let

$T: \mathbb{R}^n \mapsto \mathbb{R}^m$ be a linear transformation. Then there are some important properties of

$T$ which will be examined in this section.

2.E: Exercises

https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/02%3A_Matrices/2.E%3A_Exercises

An

$LU$ factorization of the coefficient matrix is \[\left [

$\begin{array}{rrr} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 5 & 4 \end{array}$ \right ] = \left [ \begin{array}{rrr} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 1 ...An

$LU$ factorization of the coefficient matrix is

$\left [ \begin{array}{rrr} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 5 & 4 \end{array} \right ] = \left [ \begin{array}{rrr} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 1 & 1 \end{array} \right ] \left [ \begin{array}{rrr} 1 & 2 & 3 \\ 0 & -1 & -5 \\ 0 & 0 & 0 \end{array} \right ]\nonumber$ First solve \[\left [

$\begin{array}{rrr} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 1 & 1 \end{array}$ \right ] \left [

$\begin{array}{c} u \\ v \\ w \end{array}$ \right ] =\left [ \begin{arra…

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