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**3.2 Bases and Linear Independence**

Every year Linear Independence Day is Celebrated in the US on July 4 For more information visit:

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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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**What method could we use to systematically check for independence?**

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 Rn is a subspace if it has the following 3 properties W contains the zero Vector in Rn W is closed under addition (of two vectors are in W then their sum is in W) W is closed under scalar multiplication

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**What are all of the possible vector subspaces in R2?**

What are all of the possible subspaces in R3?

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**What are all of the possible vector subspaces in R2?**

The zero vector Lines passing through the origin All of R2 What are all of the possible subspaces in R3? Planes passing through the origin All of R3

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**Examples of vector Subspaces**

A(x) is a linear transformation from Rm to Rn Is ker(A) a subspace in Rm? Is the image (A) a subspace in Rn? Is the line y = x a subspace of R2 Is the line y = x + 1 a subspace of R2

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**Examples of vector Subspaces**

Is the line y = x a subspace of R2 yes Is the line y = x + 1 a subspace of R2 No A(x) is a linear transformation from Rm to Rn Is ker(A) a subspace in Rm? yes Is the image (A) a subspace in Rn? 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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5.1 Real Vector Spaces.

5.1 Real Vector Spaces.

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