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1.3 SubspacesMath 4377/6308 Advanced Linear Algebra1.3 SubspacesJiwen HeDepartment of Mathematics, University of Houstonjiwenhe@math.uh.edumath.uh.edu/ jiwenhe/math4377Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 20151 / 26
1.3 SubspacesSubspaces1.3 SubspacesSubspaces: DefinitionSubspaces: ExamplesDetermining SubspacesZero VectorAdditive Closurescalar multiplication closureSubspaces of R2 and R3Intersections and Unions of SubspacesSums and Direct Sums of SubspacesSymmetric and Skew-symmetric MatricesJiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 20152 / 26
1.3 SubspacesSubspacesSubspacesVector spaces may be formed from subsets of other vectors spaces.These are called subspaces.Definition (Subspace)A subset W of a vector space V is called a subspace of V if W isa vector space in its own right under the operations obtained byrestricting the operations of V to W .ExampleNote that V and {0} are subspaces of any vector space V . {0} iscalled the zero subspace of V . We call these the trivialsubspaces of V .Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 20153 / 26
1.3 SubspacesSubspacesVerification of SubspacesIt is clear that properties (VS 1,2,5-8) hold for any subset ofvectors in a vector space. Therefore, a subset W of a vector spaceV is a subspace of V if and only if:1x y W whenever x, y W .2cx W whenever c F and x W3W has a zero vector4Each vector in W has an additive inverse in W .Furthermore, the zero vector of W must be the same as of V , andproperty 4 follows from property 2 and Theorem 1.2.Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 20154 / 26
1.3 SubspacesSubspacesVerification of Subspaces (cont.)Theorem (1.3)A subset W of a vector space V is a subspace of V if and only if(a) 0 W(b) x y W whenever x, y W(W is closed under vector addition)(c) cx W whenever c F and x W(W is closed under scalar multiplication).A nonempty subset W of a vector space V is a subspace of V ifand only if W is closed under addition and scalar multiplication or,equivalently, W is closed under linear combinations, that is, a, b F , u, v WJiwen He, University of Houston au bv WMath 4377/6308, Advanced Linear AlgebraSpring, 20155 / 26
1.3 SubspacesSubspacesSubspaces: ExampleExample a Let H 0 : a and b are real . Show that H is a bsubspace of R3 .Solution: Verify properties a, b and c of the definition of asubspace.a. The zero vector of R3 is in H (let a and b ).b. Adding two vectors in H always produces another vector whoseand therefore the sum of two vectors in H issecond entry isalso in H. (H is closed under addition)c. Multiplying a vector in H by a scalar produces another vector inH (H is closed under scalar multiplication).Since properties a, b, and c hold, V is a subspace of R3 .Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 20156 / 26
1.3 SubspacesSubspacesSubspaces: Example (cont.)NoteVectors (a, 0, b) in H look and act like the points (a, b) in R2 .Graphical Depiction of HJiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 20157 / 26
1.3 SubspacesSubspacesSubspaces: ExampleExample Is H x: x is real a subspace ofx 1I.e., does H satisfy properties a, b and c?Solution: For H to be a subspace of R2 , all three propertiesmust holdProperty (a) failsProperty (a) is not true becauseTherefore H is not a subspace of R2 .Jiwen He, University of HoustonMath 4377/6308, Advanced Linear Algebra.Spring, 20158 / 26
1.3 SubspacesSubspacesSubspaces: Example (cont.)Another way to show that H is not a subspace of R2 :Let u 01 and v 1, then u v 2 1and so u v , which is3and so H is not a subspace of R2 . in H. So property (b) failsProperty (b) failsJiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 20159 / 26
1.3 SubspacesSubspacesSubspaces: ExampleExample x y: 2x 5y 7z 0 a subspace of R3 ?Is W zJiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201510 / 26
1.3 SubspacesSubspacesSubspaces: Example (Zero Vector)Example xIs W : 3x 5y 12 a subspace of R2 ?y(zero vector?)Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201511 / 26
1.3 SubspacesSubspacesSubspaces: Example (Additive Closure)Example xIs W : xy 0 a subspace of R2 ?y(additive closure?)Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201512 / 26
1.3 SubspacesSubspacesSubspaces: Example (scalar multiplication closure)Example xIs W : x Z, y Z a subspace of R2 ?y(scalar multiplication closure?)Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201513 / 26
1.3 Subspaces2Subspaces of R and RSubspaces3ExampleThe subspaces of R2 consist of {0}, all lines through theorigin, and R2 itself.The subspaces of R3 are {0}, all lines through the origin, allplanes through the origin, and R3 .In fact, these exhaust all subspaces of R2 and R3 , respectively. Toprove this, we will need further tools such as the notion of basesand dimensions to be discussed soon. In particular, this shows thatlines and planes that do not pass through the origin are notsubspaces (which is not so hard to show!).Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201514 / 26
1.3 SubspacesSubspacesIntersections of SubspacesTheorem (1.4)Any intersection of subspaces of a vector space V is a subspace ofV.Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201515 / 26
1.3 SubspacesSubspacesUnions of SubspacesHowever, the union of subspaces is not necessarily a subspace,since it need not be closed under addition.Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201516 / 26
1.3 SubspacesSubspacesSums of SubspacesDefinition (Subspace Sum)Let U1 , U2 V be subspaces of a vector space V . Define the(subspace) sum of U1 and U2 to be the setU1 U2 {u1 u2 u1 U1 , u2 U2 }TheoremU1 U2 is a subspace of V . In fact, U1 U2 is the smallestsubspace of V that contains both U1 and U2 .Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201517 / 26
1.3 SubspacesSubspacesSubspace Sum: ExampleExampleLet U1 {(x, 0, 0) x R} and U2 {(0, y , 0) y R}. ThenU1 U2 {(x, y , 0) x, y R}.ExampleLet U1 {(x, 0, 0) x R} and U2 {(y , y , 0) y R}. ThenU1 U2 ?Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201518 / 26
1.3 SubspacesSubspacesSubspace Sum: ExampleExampleR2 U U 0 .The union U U 0 of two subspaces is not necessarily a subspace.Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201519 / 26
1.3 SubspacesSubspacesDirect Sums of SubspacesDefinition (Direct Sum of Subspaces)Suppose every u U can be uniquely written as u u1 u2 foru1 U1 and u2 U2 . Then we useU U1 U2to denote the direct sum of U1 and U2 .Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201520 / 26
1.3 SubspacesSubspacesDirect Sum: ExampleExampleLet U1 {(x, y , 0) x, y R} and U2 {(0, 0, z) z R}. ThenR3 U1 U2 .ExampleLet U1 {(x, y , 0) x, y R} and U2 {(0, w , z) w , z R}.ThenR3 U1 U2 , but is not the direct sum of U1 and U2 .Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201521 / 26
1.3 SubspacesSubspacesVerification of Direct SumProposition 1Let U1 , U2 V be subspaces. Then V U1 U2 if and only ifthe following two conditions hold:1V U1 U2 ;2If 0 u1 u2 with u1 U1 and u2 U2 , then u1 u2 0.Proposition 2Let U1 , U2 V be subspaces. Then V U1 U2 if and only ifthe following two conditions hold:1V U1 U2 ;2U1 U2 {0};Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201522 / 26
1.3 SubspacesSubspacesDirect Sum: ExampleExampleLetU1 {p P2n a0 a2 t 2 · · · a2n t 2n },U2 {p P2n a1 t a3 t 3 · · · a2n 1 t 2n 1 }.ThenP2n U1 U2 .Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201523 / 26
1.3 SubspacesSubspacesSymmetric MatricesExampleThe transpose At of an m n matrix A is the n m matrixobtained by interchanging rows and columns of A, that is,(At )ij Aji .A symmetric matrix A has At A and must be square.The set Sym of all symmetric matrices in Mn (i.e. Mn n (F ))is a subspace of Mn .ExampleAn n n matrix A is a diagonal matrix if Aij 0 wheneveri 6 jThe set of diagonal matrices is a subspace of Mn .Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201524 / 26
1.3 SubspacesSubspacesSkew-Symmetric MatricesExampleA skew-symmetric matrix A has At A and must besquare.The set SkewSym of all skew-symmetric matrices in Mn is asubspace of Mn .Jiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201525 / 26
1.3 SubspacesSubspacesDirect Sum: Symmetric and Skew-Symmetric MatricesAny matrix A can be written in the form11A (A At ) (A At ) B C .22It is easy to verify that B is symmetric and C is skew-symmetricand so we have a decomposition of A as the sum of a symmetricmatrix and a skew-symmetric matrix.Since the sets Sym and SkewSym of all symmetric andskew-symmetric matrices in Mn are subspaces of Mn , we haveMn Sym SkewSymSince Sym SkewSym {0}, we haveMn Sym SkewSymJiwen He, University of HoustonMath 4377/6308, Advanced Linear AlgebraSpring, 201526 / 26
Jiwen He, University of Houston Math 4377/6308, Advanced Linear Algebra Spring, 2015 24 / 26. 1.3 Subspaces Subspaces Skew-Symmetric Matrices Example A skew-symmetric matrix A has At A and must be sq
