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3.2 Bases and Linear Independence For more information visit: a-Vector-Space.topicArticleId-20807,articleId html Every year Linear Independence Day is Celebrated in the US on July 4

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Independence A set of vectors is said to be (Linearly) independent if no vector is a linear combination of the other vectors.

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Problem 10 Determine if the vectors are linearly independent

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Problem 10 Solution

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Problem 16 Determine if the vectors are Linearly Independent or dependent.

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Problem 16 solution

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What method could we use to systematically check for independence?

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Write the vectors as the columns of a matrix. Find rref(A) The columns with leading ones are independent. The vectors with out leading ones are dependent.

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Subspace A subset W in R n is a subspace if it has the following 3 properties a)W contains the zero Vector in R n b)W is closed under addition (of two vectors are in W then their sum is in W) c)W is closed under scalar multiplication

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What are all of the possible vector subspaces in R 2 ? What are all of the possible subspaces in R 3 ?

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What are all of the possible vector subspaces in R 2 ? The zero vector Lines passing through the origin All of R 2 What are all of the possible subspaces in R 3 ? The zero vector Lines passing through the origin Planes passing through the origin All of R 3

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Examples of vector Subspaces A(x) is a linear transformation from R m to R n Is ker(A) a subspace in R m ? Is the image (A) a subspace in R n ? Is the line y = x a subspace of R 2 Is the line y = x + 1 a subspace of R 2

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Examples of vector Subspaces Is the line y = x a subspace of R 2 yes Is the line y = x + 1 a subspace of R 2 No A(x) is a linear transformation from R m to R n Is ker(A) a subspace in R m ? yes Is the image (A) a subspace in R n ? yes

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Basis A set of vectors is a basis of a subspace if the vectors are independent and they span the subspace.

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Problem 28 Find a basis of the image (column space)

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Problem 28 solution

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Example 4 Consider the matrix Column 2 is a multiple of column 1 and therefore adds no new information about the column space (Image) Column 4 is a sum of columns 1 and 3 therefore it also gives no new information about the column space (Image)

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Example 4 The image of A can be spanned by 2 vectors but not by 1 alone. (using 3 vectors would be redundant) Therefore say the column vectors from the first and third column of the matrix form a basis of the subspace

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Homework p odd

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