INTRODUCTION TO LIE ALGEBRAS. LECTURE 2. 2. More
... Lemma 2.5.4. Let f ∈ End(V ) be diagonalizable and let W be a f invariant subspace of V (i.e., f (W ) ⊆ W ). Then W = ⊕λ∈S Wλ where Wλ = W ∩ Vλ . P Proof. We have to prove that if x ∈ W and if x = xλ with xλ ∈ Vλ then xλ ∈ W . P P In fact, W 3 f k (x) = f k (xλ ) = λk xλ for each k. Let T = {λ ∈ S|x ...
... Lemma 2.5.4. Let f ∈ End(V ) be diagonalizable and let W be a f invariant subspace of V (i.e., f (W ) ⊆ W ). Then W = ⊕λ∈S Wλ where Wλ = W ∩ Vλ . P Proof. We have to prove that if x ∈ W and if x = xλ with xλ ∈ Vλ then xλ ∈ W . P P In fact, W 3 f k (x) = f k (xλ ) = λk xλ for each k. Let T = {λ ∈ S|x ...
(pdf)
... 16. Suppose that A is a 6 × 6 matrix with characteristic polynomial cA (λ) = (1 + λ)(1 − λ)2 (2 − λ)3 . (a) Prove that it is impossible to find three linearly independent vectors vi , i = 1, 2, 3, such that Avi = vi , i = 1, 2, 3. ...
... 16. Suppose that A is a 6 × 6 matrix with characteristic polynomial cA (λ) = (1 + λ)(1 − λ)2 (2 − λ)3 . (a) Prove that it is impossible to find three linearly independent vectors vi , i = 1, 2, 3, such that Avi = vi , i = 1, 2, 3. ...
Linear Transformations
... Definition 5. Let Matm×n (F ) be the set of all m × n matrices with elements in F . Let + : Mat(F ) × Mat(F ) → Mat(F ) and α : F × Mat(F ) → Mat(F ); (Matm×n (F ), +, α) is a vector space. With this definition, the following theorem is presented without proof. Theorem 3. Let Hom(F n , F m ) be the ...
... Definition 5. Let Matm×n (F ) be the set of all m × n matrices with elements in F . Let + : Mat(F ) × Mat(F ) → Mat(F ) and α : F × Mat(F ) → Mat(F ); (Matm×n (F ), +, α) is a vector space. With this definition, the following theorem is presented without proof. Theorem 3. Let Hom(F n , F m ) be the ...
MTH6140 Linear Algebra II 6 Quadratic forms ∑ ∑
... same function Q. The effect of a change of basis is a linear substitution v = Pv0 on the variables, where P is the transition matrix between the bases. Thus we have v> Av = (Pv0 )> A(Pv0 ) = (v0 )> (P> AP)v0 , so we have the following: Proposition 6.1 A basis change with transition matrix P replaces ...
... same function Q. The effect of a change of basis is a linear substitution v = Pv0 on the variables, where P is the transition matrix between the bases. Thus we have v> Av = (Pv0 )> A(Pv0 ) = (v0 )> (P> AP)v0 , so we have the following: Proposition 6.1 A basis change with transition matrix P replaces ...
Free associative algebras
... Here’s why it’s not necessary to define n-fold tensor products separately. Suppose n = p + q, with p and q both positive integers. It’s easy to see that an n-linear map from U1 × · · · × Un to W is the same thing as a bilinear map Mult(U1 , . . . , Up ; W ) × Mult(Up+1 , . . . , Up+q ; W ). (0.6b) T ...
... Here’s why it’s not necessary to define n-fold tensor products separately. Suppose n = p + q, with p and q both positive integers. It’s easy to see that an n-linear map from U1 × · · · × Un to W is the same thing as a bilinear map Mult(U1 , . . . , Up ; W ) × Mult(Up+1 , . . . , Up+q ; W ). (0.6b) T ...
10-31-16-graphing-linear-equations-real
... What is a Linear Equation? YIP: Linear equations are usually shown on a coordinate plane YIP: Real life situations of linear equations: - the stock market - the payments of a car - Sports statistics ...
... What is a Linear Equation? YIP: Linear equations are usually shown on a coordinate plane YIP: Real life situations of linear equations: - the stock market - the payments of a car - Sports statistics ...
V. Topological vector spaces
... If F = R, then A is absolutely convex, if and only if it is convex and symmetric. co(A) = {t1 x1 + · · · + tk xk ; x1 , . . . , xk ∈ A, t1 , . . . , tk ≥ 0, t1 + · · · + tk = 1}. b(A) = {αx; x ∈ A, α ∈ F, |α| ≤ 1}. aco(A) = co(b(A)). ...
... If F = R, then A is absolutely convex, if and only if it is convex and symmetric. co(A) = {t1 x1 + · · · + tk xk ; x1 , . . . , xk ∈ A, t1 , . . . , tk ≥ 0, t1 + · · · + tk = 1}. b(A) = {αx; x ∈ A, α ∈ F, |α| ≤ 1}. aco(A) = co(b(A)). ...
Exam 1 solutions
... 9.(10pts) Can a square matrix with two identical columns be invertible? (Explain). The Invertible Matrix Theorem tells us that if a square matrix is invertible, then its columns must be linearly independent. If two of the columns are the same, then the columns are clearly dependent. We conclude that ...
... 9.(10pts) Can a square matrix with two identical columns be invertible? (Explain). The Invertible Matrix Theorem tells us that if a square matrix is invertible, then its columns must be linearly independent. If two of the columns are the same, then the columns are clearly dependent. We conclude that ...
Linear Space - El Camino College
... The terms linear space and vector space mean the same thing and can be used interchangeably. I have used the term linear space in the discussion below because I prefer it, but that is a personal preference. To start with, a linear space consists of a non-empty set, and 2 operations, addition, and mu ...
... The terms linear space and vector space mean the same thing and can be used interchangeably. I have used the term linear space in the discussion below because I prefer it, but that is a personal preference. To start with, a linear space consists of a non-empty set, and 2 operations, addition, and mu ...
Basis (linear algebra)
Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.A set of vectors in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set. In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space V, every element of V can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.