A x = x_{1} a_{1} + \dots + x_{4} a_{4}

33:["This is the Consistency Theorem (1.3.1). Writing ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"$21a"}}],", solving the system ","$L21b"," asking whether ","$L21c"," can be built from the columns."] 34:[["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

b

"}}]," is a linear combination of the rows of ",["$","span","1",{"dangerouslySetInnerHTML":{"__html":"

A

"}}]] 35:["Wrong space entirely: ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

b \in R 3

"}}]," while the rows live in ",["$","span","1",{"dangerouslySetInnerHTML":{"__html":"

R 1 \times 4

"}}],". ",["$","span","2",{"dangerouslySetInnerHTML":{"__html":"

A x

"}}]," is a combination of ",["$","em","3",{"className":"italic","children":["columns"]}],"."] 36:[["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A

"}}]," is nonsingular"] 37:[["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A

"}}]," isn't even square, so 'nonsingular' doesn't apply. Only square matrices are called singular/nonsingular."] 38:[["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

b = 0

"}}]] 39:["That guarantees consistency but is far from necessary — plenty of nonzero ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

b

"}}]," lie in the column space."] 3a:[["If ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A

"}}]," is ",["$","span","1",{"dangerouslySetInnerHTML":{"__html":"

3 \times 5

"}}]," and ",["$","span","2",{"dangerouslySetInnerHTML":{"__html":"

B

"}}]," is ",["$","span","3",{"dangerouslySetInnerHTML":{"__html":"

5 \times 3

"}}],", then both ",["$","span","4",{"dangerouslySetInnerHTML":{"__html":"

A B

"}}]," and ",["$","span","5",{"dangerouslySetInnerHTML":{"__html":"

B A

"}}]," are defined and both are ",["$","span","6",{"dangerouslySetInnerHTML":{"__html":"

3 \times 3

"}}],"."],null] 3b:["$","span",null,{"children":"True"}] 21d:T406,

A B \neq = B A

3c:["Both products are defined, but ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A B

"}}]," is ",["$","span","1",{"dangerouslySetInnerHTML":{"__html":"

3 \times 3

"}}]," while ",["$","span","2",{"dangerouslySetInnerHTML":{"__html":"

B A

"}}]," is ",["$","span","3",{"dangerouslySetInnerHTML":{"__html":"

5 \times 5

"}}],". Inner dimensions must match; the ",["$","em","4",{"className":"italic","children":["outer"]}]," dimensions give the shape. This is the cleanest illustration that ",["$","span","5",{"dangerouslySetInnerHTML":{"__html":"$21d"}}]," in general — they need not even be the same size."] 3d:["$","span",null,{"children":"False"}] 3e:[["If ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A

"}}]," is any ",["$","span","1",{"dangerouslySetInnerHTML":{"__html":"

m \times n

"}}]," matrix, then ",["$","span","2",{"dangerouslySetInnerHTML":{"__html":"

A T A

"}}]," is symmetric."],null] 3f:["$","span",null,{"children":"True"}] 21e:Td84,

(A^{T} A)^{T} = A^{T} (A^{T})^{T} = A^{T} A

40:[["$","span","0",{"dangerouslySetInnerHTML":{"__html":"$21e"}}],", using ","$L21f",". This shows up again in Chapter 5 — it's why the normal equations ","$L220"," have such nice structure."] 41:["$","span",null,{"children":"False"}] 42:[["If ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A

"}}]," and ",["$","span","1",{"dangerouslySetInnerHTML":{"__html":"

B

"}}]," are nonsingular ",["$","span","2",{"dangerouslySetInnerHTML":{"__html":"

n \times n

"}}]," matrices, then ",["$","span","3",{"dangerouslySetInnerHTML":{"__html":"

(A B) - 1

"}}]," equals:"],null] 221:T473,

A^{- 1} B^{- 1}

43:[["$","span","0",{"dangerouslySetInnerHTML":{"__html":"$221"}}]] 222:Tad8,

(A B) (A^{- 1} B^{- 1}) = A (B A^{- 1}) B^{- 1}

223:T5c9,

B A^{- 1} = A^{- 1} B

44:["Check it: ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"$222"}}],", and there is no reason ",["$","span","1",{"dangerouslySetInnerHTML":{"__html":"$223"}}],". The order must reverse."] 224:T473,

B^{- 1} A^{- 1}

45:[["$","span","0",{"dangerouslySetInnerHTML":{"__html":"$224"}}]] 225:Tf34,

(A B) (B^{- 1} A^{- 1}) = A (B B^{- 1}) A^{- 1} = A I A^{- 1} = I

46:["Theorem 1.4.2. Verify: ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"$225"}}],". Inverting a product reverses the order — same pattern as the transpose rule ","$L226","."] 47:[["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

(B A) - 1

"}}]] 227:T7cb,

(B A)^{- 1} = A^{- 1} B^{- 1}

228:T473,

B^{- 1} A^{- 1}

48:[["$","span","0",{"dangerouslySetInnerHTML":{"__html":"$227"}}],", which is generally different from ",["$","span","1",{"dangerouslySetInnerHTML":{"__html":"$228"}}],"."] 49:[["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

B A - 1

"}}]] 4a:["Dimensionally fine but algebraically wrong — this isn't even an inverse in general."] 4b:[["If ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A B = O

"}}]," for ",["$","span","1",{"dangerouslySetInnerHTML":{"__html":"

n \times n

"}}]," matrices, then ",["$","span","2",{"dangerouslySetInnerHTML":{"__html":"

A = O

"}}]," or ",["$","span","3",{"dangerouslySetInnerHTML":{"__html":"

B = O

"}}],"."],null] 4c:["$","span",null,{"children":"True"}] 229:T78c,

A = (1010)

22a:T7e5,

B = (1 - 1 1 - 1)

4d:["Matrices have zero divisors. Take ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"$229"}}],", ",["$","span","1",{"dangerouslySetInnerHTML":{"__html":"$22a"}}],": neither is zero yet ","$L22b",". What ","$L22c"," true: if ","$L22d"," is nonsingular and ","$L22e",", then multiplying by ","$L22f"," forces ","$L230","."] 4e:["$","span",null,{"children":"False"}] 4f:[["If ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A B = A C

"}}]," and ",["$","span","1",{"dangerouslySetInnerHTML":{"__html":"

A \neq = O

"}}],", then ",["$","span","2",{"dangerouslySetInnerHTML":{"__html":"

B = C

"}}],"."],null] 50:["$","span",null,{"children":"True"}] 51:["Cancellation needs ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A

"}}]," to be ",["$","strong","1",{"className":"text-ink font-semibold","children":["nonsingular"]}],", not merely nonzero. ",["$","span","2",{"dangerouslySetInnerHTML":{"__html":"

A B = A C

"}}]," gives ",["$","span","3",{"dangerouslySetInnerHTML":{"__html":"

A (B - C) = O

"}}],"; if ",["$","span","4",{"dangerouslySetInnerHTML":{"__html":"

A

"}}]," is singular there are nonzero matrices in that null space. With ",["$","span","5",{"dangerouslySetInnerHTML":{"__html":"

A - 1

"}}]," available, multiply on the left and you do get ",["$","span","6",{"dangerouslySetInnerHTML":{"__html":"

B = C

"}}],"."] 52:["$","span",null,{"children":"False"}] 53:[["You multiply a ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

3 \times 3

"}}]," matrix ",["$","span","1",{"dangerouslySetInnerHTML":{"__html":"

A

"}}]," on the ",["$","strong","2",{"className":"text-ink font-semibold","children":["right"]}]," by the elementary matrix formed by swapping rows 1 and 2 of ",["$","span","3",{"dangerouslySetInnerHTML":{"__html":"

I

"}}],". The result is:"],null] 54:[["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A

"}}]," with rows 1 and 2 swapped"] 55:["That's left multiplication, ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

E A

"}}],". Right multiplication acts on columns."] 56:[["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A

"}}]," with columns 1 and 2 swapped"] 57:["Premultiplying by ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

E

"}}]," performs the row operation; postmultiplying performs the corresponding ",["$","strong","1",{"className":"text-ink font-semibold","children":["column"]}]," operation. So ",["$","span","2",{"dangerouslySetInnerHTML":{"__html":"

A E

"}}]," swaps columns 1 and 2."] 58:[["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A

"}}]," unchanged"] 59:["Only if ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

E = I

"}}],", which it isn't."] 5a:[["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A T

"}}]," with rows 1 and 2 swapped"] 5b:["No transpose is involved in ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A E

"}}],"."] 5c:[["An ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

n \times n

"}}]," matrix ",["$","span","1",{"dangerouslySetInnerHTML":{"__html":"

A

"}}]," is nonsingular if and only if ",["$","span","2",{"dangerouslySetInnerHTML":{"__html":"

A

"}}]," is row equivalent to ",["$","span","3",{"dangerouslySetInnerHTML":{"__html":"

I

"}}],"."],null] 5d:["$","span",null,{"children":"True"}] 5e:["Theorem 1.5.2 — these are equivalent, along with '",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"

A x = 0

"}}]," has only the trivial solution'. It's also the engine behind computing ",["$","span","1",{"dangerouslySetInnerHTML":{"__html":"

A - 1

"}}],": row reduce ",["$","span","2",{"dangerouslySetInnerHTML":{"__html":"

(A ∣ I)

"}}]," and if the left half becomes ",["$","span","3",{"dangerouslySetInnerHTML":{"__html":"

I

"}}],", the right half is ",["$","span","4",{"dangerouslySetInnerHTML":{"__html":"

A - 1

"}}],"."] 5f:["$","span",null,{"children":"False"}] 60:[["Every upper triangular matrix is nonsingular."],null] 61:["$","span",null,{"children":"True"}] 231:T67b,

(1020)

32c:I[27201,["/_next/static/immutable/chunks/1ucnsl9s5r8im.js"],"IconMark"] 62:["A triangular matrix may have zeros on the diagonal — e.g. ",["$","span","0",{"dangerouslySetInnerHTML":{"__html":"$231"}}]," is upper triangular and singular. For the coefficient matrix of a ",["$","em","1",{"className":"italic","children":["strict triangular system"]}]," you additionally need all diagonal entries nonzero."] 63:["$","span",null,{"children":"False"}] 232:T14e4,

0002241035010033

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