Linear Algebra and Its Applications (4th Ed) David Lay - CONDENSED (Only Formulas/Definitions) PDF

Preview

Summary

These is a condensed version of the book up to Chapter 4.6. It contains only the formulas/theorems and major definitions for each chapter....

Description

LINEAR ALGEBRA – EXAM 2 STUDY GUIDE Pre-Knowledge (in chapter one – not on the test, but need to know) basic variable – variables that correspond to the pivot columns. free variable – variables that are not associated with the pivot columns.

Figure 1. Is the range of T all of 𝑅 𝑚 ?

Figure 2. Is every b the image of at most one vector?

Section 2.1: MATRIX OPERATIONS Diagonal entries – in an 𝑚 𝑥 𝑛 matrix A, are 𝑎11 , 𝑎22 , 𝑎33 , …. they form the main diagonal. Diagonal matrix – a square 𝑛 𝑥 𝑛 matrix whose nondiagonal matrices entries are zero.

or identity matrix -> Figure 3. Examples of diagonal matrices

Zero matrix – an 𝑚 𝑥 𝑛 matrix whose entries are all zero. The size of zero matrix is usually determined by context.

Equal – two matrices are equal if they have the same size (rows, columns) and the entries are the same. sum of A and B; (A + B) – sum of the corresponding entries. 𝑎11 + 𝑏11 , 𝑒𝑡𝑐. OPERATION IS ONLY VALID IF A AND B ARE THE SAME EXACT SIZE. 𝑎11 + 𝑏11 ⋮ [ 𝑎𝑚1 + 𝑏𝑚1

⋯ 𝑎1𝑛 + 𝑏1𝑛 ⋱ ⋮ ] ⋯ 𝑎𝑚𝑛 + 𝑏𝑚𝑛

Figure 4. shows a summary of matrix addition, A + B

Scalar multiple – is the resulting matrix 𝑟𝐴 where 𝑟 is a scalar and 𝐴 is a matrix

Figure 5. showing that (AB)x is equivalent to A(Bx)

This is one way to compute the product. The other goes straight to the answer.

NOTE: Multiplication of matrices corresponds to composition of linear transformations. For matrix multiplication to be valid…. the inner need to match – result is the outer.

Figure 6. demonstration if matrix multiplication is valid for the matrices and shows the resulting matrix product size\

The direct to answer solution is…

Figure 7. demonstrating Theorem 2(d): 𝐼𝑚 𝐴 = 𝐴 = 𝐴𝐼𝑛

Powers of a Matrix - 𝐴𝑘 denotes the product of k copies of A if A is an 𝑛 𝑥 𝑛 (Square!) and 𝑘 is a positive integer.

Figure 8. shows the powers of a matrix

Note: 𝐴0 can be interpreted as the identity matrix.

transpose of A; (𝐴𝑇 ) – given 𝑚 x 𝑛 matrix 𝐴, is the 𝑛 x 𝑚 matrix whose columns are formed from the corresponding rows of A; rows become columns, columns become rows.

Figure 9. Examples of transpose of a matrix A

Theorem 3(d), (𝐴𝐵)𝑇 = 𝐵𝑇 𝐴𝑇 can be expanded…

THUS, (𝐴𝐵𝐶)𝑇 = 𝐶𝑇 𝐵𝑇 𝐴𝑇

(𝐴𝐵𝐶𝐷)𝑇 = 𝐷𝑇 𝐶 𝑇 𝐵𝑇 𝐴𝑇 … and on and on…

Section 2.2: THE INVERSE OF A MATRIX Invertible – An 𝑛 x 𝑛 (only square matrices are invertible!!) matrix 𝐴 is said to be invertible if there is an 𝑛 x 𝑛 matrix C such that 𝐶𝐴 = 𝐼 and 𝐴𝐶 = 𝐼

in this case C is the inverse of A. The unique inverse is denoted by 𝐴−1 , so that

𝐴−1 𝐴 = 𝐼 and 𝐴𝐴−1 = 𝐼

singular matrix – a matrix that is NOT invertible nonsingular matrix – matrix that is invertible

put another way..

𝐴−1 =

For 2 x 2 matrix

If det(𝐴) = 0, then A is not invertible => A is singular

1 𝑑 [ det(𝐴) −𝑐

−𝑏 ] 𝑎

Theorem 3(d), (𝐴𝐵)−1 = 𝐵 −1 𝐴−1 can be expanded…

(𝐴𝐵𝐶)−1 = 𝐶 −1 𝐵−1 𝐴−1

(𝐴𝐵𝐶𝐷)−1 = 𝐷−1 𝐶 −1 𝐵−1 𝐴−1 … and on and on… Elementary matrix – matrix that is obtained by performing a single elementary operation on an identity matrix. Examples:

Section 2.3: Characteristics of Invertible Matrices

WARNING: The Invertible Matrix Theorem only applies to square matrices.

Figure 10. 𝐴−1 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑠 𝐴𝑥 𝑏𝑎𝑐𝑘 𝑡𝑜 𝑥

Section 2.8: Subspaces of ℝ𝑛

Figure 11. The standard basis for ℝ3

Section 2.9: Dimension and Rank

Section 3.1: Introduction to Determinants

Section 3.2: Properties of Determinants

Section 4.1: Vector Spaces and Subspaces

NOTE: A subspace is a vector space itself.

Section 4.2: Null Spaces, Columns Spaces, and Linear Transformations

Section 4.3: Linearly Independent Sets; Bases

Section 4.5: The Dimension of a Vector Space

Section 4.6: Rank...