2331_Notes_4o1_fill - Math 2331 Linear Algebra Section 4.1...

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Math 2331 Linear Algebra Section 4.1 Orthogonal Subspaces Two vectors are said to be orthogonal if their dot product is 0. This translates the concept of perpendicular vectors in 2- and 3-dimensional space into a more general setting. Definition - Given a vector space, two SUBSPACES V and W are orthogonal if every vector in V is orthogonal to every vector in W. In other words, V and W are orthogonal if 0 vw •= for every v in V and every w in W. Let's consider orthogonal subspaces of The plane - 3-D Space -
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Subspaces of 3-D space that are NOT orthogonal, but we might think they are - Consider two adjacent walls - Orthogonality and the 4 Fundamental Subspaces of A 1. The Row Space and the Null Space -
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Example - 10 1 01 3 00 0 A ⎛⎞ ⎜⎟ = ⎝⎠ Null Space - Row Space - Row dot Null vectors - Column Space - Left null space
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Column dot left null space Another Example 12 4 31 24 6 42 36 1 051 A ⎛⎞ =− ⎜⎟ ⎝⎠ RREF 12002 00101 00011 R = Column Space - Null Space of - left null space T A
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Try the dot product of two vectors - Row Space - Null Space - Dot Product -
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2331_Notes_4o1_fill - Math 2331 Linear Algebra Section 4.1...

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