2.1: matrix operations - ndsu - north …stiszler/129ch2.pdf55 2.3: characterizations of invertible...
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2.1: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are 2 matrices equal? Scalar Multiplication
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Matrix Addition
Theorem 1 (pg 130)
Let A, B, and C be matrices of the same size, and let r & s be scalars.
1.) ABBA +=+
2.) C)(BACB)(A ++=++
3.) A0A =+
4.) rBrAB)r(A +=+
5.) sArAs)A(r +=+
6.) (rs)Ar(sA) =
Example 1
Let
−=
01
32
46
A and
−
−
=
42
15
38
B . Compute each of the following:
=2A-
=2A-B
=+ 2BA
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MATRIX MULTIPLICATION How do I multiply matrices? Row-Column Rule For Computing AB
Theorem 2 (pg 134)
Let A be an mxn matrix, and let B & C have sizes for which the indicated sums and products are defined. Let r be any scalar.
1.) (AB)CA(BC) =
2.) ACABC)A(B +=+
3.) CABAC)A(B +=+
4.) A(rB)(rA)Br(AB) ==
5.) nm AAA II ==
Associative law of multiplication
Left distributive law
Right distributive law
Identity for matrix multiplication
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Example 2
Let
−
−=
254
102A ,
−=
12
21B , and
−=
41
53C .
Compute each of the following:
=AC
=CA
=BC
=CB
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WARNINGS!
1.) In general, BAAB ≠ .
2.) The cancellation laws do NOT hold for matrix multiplication. That is, if ACAB = then it is
NOT necessarily true that CB = .
3.) If a product AB is the zero matrix, you CANNOT conclude in general that 0A = or 0B = .
What is the transpose of a matrix?
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Theorem 3 (pg 136)
Let A and B denote matrices whose sizes are appropriate for the following sums and products and let r be any scalar.
1.) ( ) AATT =
2.) ( ) TTTBABA +=+
3.) ( ) TTrArA =
4.) ( ) TTTABAB =
Example 3
Let
=
43
21A and
=
6
5x . Compute each of the following:
( ) =TAx
=TTAx
=Txx
=xxT
Is TTxA defined? Why or why not.
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2.2: THE INVERSE OF A MATRIX When is a matrix A invertible? Singular Matrix vs. Nonsingular Matrix Theorem 4 (CAUTION: THIS ONLY WORKS FOR 2x2 MATRICES!)
Example 1
Find the inverse (if it exists) of each of the following matrices.
34
68
86
42
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Theorem 5 Theorem 6 Theorem 7
Algorithm for Finding 1A − (pg 153)
1.) Set up the augmented matrix [A I].
2.) Row reduce the matrix into reduced echelon form.
3.) If A is row equivalent to I, then [A I] is row equivalent to [I 1A − ]. Otherwise, A does not
have an inverse.
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Example 2
Let
=
801
352
321
A and
=
5
12
42
b . Find 1A − and use it to solve bxA = .
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Example 3
Let A, B, C, D, X, and Y be invertible nxn matrices. Solve the equation, ( ) YDCXBA =+ for X.
Things to keep in mind: Matrix division does not exist. You cannot divide by a matrix. Also, keep
the order of multiplication consistent. If you multiply by 1A − , on the left of the left side of the
equation you must multiply by 1A − on the left of the right side of the equation as well.
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2.3: CHARACTERIZATIONS OF INVERTIBLE MATRICES
Theorem 8: The Invertible Matrix Theorem (pg 163)
Let A be a square nxn matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false.
1.) A is an invertible matrix.
2.) A is row equivalent to the nxn identity matrix, nI .
3.) A has n pivot positions.
4.) The equation 0xA = has only the trivial solution.
5.) The columns of A form a linearly independent set.
6.) The linear transformation nn RR:T → given by ( ) xAxT = is one-to-one.
7.) The equation bxA = has exactly one solution for each b in nR .
8.) The columns of A span nR .
9.) The linear transformation nn RR:T → given by ( ) xAxT = is onto.
10.) There is an nxn matrix C such that nACCA I== .
11.) TA is an invertible matrix.
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What is an invertible transformation? Theorem 9
What if nn RR:T → is one-to-one? Onto?
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Example 1
Let 33 RR:T → by
+−
−+
−
=
31
321
32
3
2
1
x4x6
xx3x2
x3x
x
x
x
T be a linear transformation. Show that T is
invertible and find 1T− .
2.4 SUBSPACES OF nR
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What is a vector space? What is a subspace? What is the subspace test?
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Example 1 Determine which of the following are subspaces of R2.
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Example 2
Use the subspace test to determine if the following is a subspace of 3R .
∈
+
−
+
= Ry,x
y4x3
xy
y2x
H
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Example 3
Use the subspace test to determine if the following is a subspace of 3R .
∈
−
+
= Rc,b,a
cb
a
cb
W 2
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What is ColA? What is NulA? Theorem 10
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Example 4
Let
−
−
−
−−
=
075
387
312
121
A .
• What is ColA?
• ColA is a subspace of kR , what is k in this example?
• What is NulA?
• NulA is a subspace of sR , what is s in this example?
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What is a basis?
What is the standard basis for nR ? Theorem 11
Example 5
Let
−
−
−
−−
=
075
387
312
121
A . Find a basis for ColA and NulA.
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Example 6 Determine which sets are bases for R2 or R3. Justify each answer.
3
1,
−1
1
2
1,
− 3
2,
2
3
1
1
1
,
3
2
1
,
0
1
0
3
2
1
,
3
1
2
,
15
7
8
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2.5 DIMENSION & RANK What is the dimension of a subspace?
Example 1
Determine the dimension of the subspace H of 3R spanned by the vectors
=
2
2
1
v1 ,
=
1
2
3
v 2 ,
=
7
10
11
v 3 ,
=
4
6
7
v 4 .
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What is the rank of a matrix? Theorem 12 (The Rank Theorem)
Example 2 Suppose a 3x5 matrix A has 3 pivot columns.
Is 3RColA = ?
Is 2RNulA = ?
Suppose a 4x7 matrix A has 3 pivot columns.
Is 3RColA = ?
What is the dimension of NulA?
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Example 3 Construct a 4x3 matrix with rank 1.
Theorem 13 (The Basis Theorem)
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Example 4
Let
−
−−
−
−
−
=
42113
52221
14132
32310
13121
A . Is the set
−
=
2
1
3
3
0
,
1
1
2
0
1
,
9
0
9
3
6
S a basis for ColA?
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Theorem 8: The Invertible Matrix Theorem (pg 190)
Let A be a square nxn matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false. 1.) A is an invertible matrix.
2.) A is row equivalent to the nxn identity matrix, nI .
3.) A has n pivot positions.
4.) The equation 0xA = has only the trivial solution.
5.) The columns of A form a linearly independent set.
6.) The linear transformation nn RR:T → given by ( ) xAxT = is one-to-one.
7.) The equation bxA = has exactly one solution for each b in nR .
8.) The columns of A span nR .
9.) The linear transformation nn RR:T → given by ( ) xAxT = is onto.
10.) There is an nxn matrix C such that nACCA I== .
11.) TA is an invertible matrix.
12.) The columns of A form a basis for nR .
13.) nRColA =
14.) ( ) nColAdim =
15.) ( ) nArank =
16.) }0{NulA =
17.) ( ) 0NulAdim =
18.) 0Adet ≠