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Vector Spaces
Since every element of W1 is orthogonal to every element of W2 is suffices to show that these two elements are linearly independent. By picking arbitrary non-zero vectors from the subspaces it can be shown that they are indeed linearly independent.
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Orthogonal subspaces
26784 Orthogonal Subspaces Let A be an mxn matrix. show that
1) If x Є N(A^TA), then Ax is in both R(A) and N(A^T).
2) N(A^TA) = N(A.)
3) A and A^TA have the same rank.
4) If A has linearly independent columns, then A^TA is nonsingular
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Continuous, Real Valued, Linearly Dependent Functions and Matrix Determinants
Part 2: proof of "if"
Here we prove that if det( ) = 0 the set of functions {f_1, ...f_k} is linearly dependent.
We do it from negation, proving that if {f_1, ...f_k} is linearly independent,
then det( ) not= 0.
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Linearly Independent Vectors and Invertible Matrices
,A(vk)} are linearly independent?
Also, if you know that {v1, v2,...,vk} are linearly independent and {B(v1), B(v2,)...,B(vk)} are linearly independent, how do you prove that B is invertible?
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Fibonacci Numbers and Golden Rule
Since the above solutions yield two linearly independent functions of type (2), their sum will be a solution of the Fibonacci equation as well:
f_n = C_1*r^n_1 + C_2*r^n_2
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Matrices and Vectors : Linear Independence
, ,
To determine whether the three vectors are linearly independent or not, we need to solve the follow equations: . This implies
So we compute
Therefore, , and are linearly independent.
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Matrices : Finding the Rank
Rank of a matrix is equal to the number of linearly independent rows in the matrix
Notice, (1) the number of linearly independent rows is always equal to the number of linearly independent rows;
(2) columns (rows) are linearly dependent if one of
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Minimal linearly dependent sets of columns
Since all the subsets of 3 column vectors of B are linearly independent, it follows that the minimal linearly dependent set of columns of B is the set {c_1,c_2,c_3,c_4 }.
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Vector Spaces : Rank
independent rows in the matrix
Notice, (1) the number of linearly independent rows is always equal to the number of linearly independent rows;
2) columns (rows) are linearly dependent if one of the columns (row) can be written as a linear combination
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Linearly Independent Subsets : Partially Ordered by Inclusion
75952 Linearly Independent Subsets : Partially Ordered by Inclusion Let X be any vector space over the field F, let L be a linearly independent subset
of X, and A be the set of linearly independent subsets of X containing L.