commit a015eaab33091aa5b658be92d426345c8a07565b
parent 8fe300af4e072eb58af851927f2c950378141792
Author: Andrew <andrewlaack1@gmail.com>
Date: Thu, 25 Jul 2024 14:33:50 -0500
Completed this stuff
Diffstat:
4 files changed, 35 insertions(+), 0 deletions(-)
diff --git a/Kernel.md b/Kernel.md
@@ -0,0 +1,12 @@
+:lin-alg:
+# Kernel
+
+Khan
+
+## Notes
+
+**Definition:** The kernel of a linear transformation is the set of all vectors that are equal to the null vector under the L.T.
+
+This is stated as ker(T), spoken as the kernel of T.
+
+This is similar to the [[NullSpace.md]] except it is specific to linear transformations.
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -47,3 +47,5 @@ Khan Unit 2:
- [[LinearTransformation.md]]
- [[IdentityMatrix.md]]
- [[Image.md]]
+ - [[Preimage.md]]
+ - [[Kernel.md]]
diff --git a/Preimage.md b/Preimage.md
@@ -0,0 +1,20 @@
+:lin-alg:
+# Preimage
+
+Khan Unit 2
+
+## Notes
+
+**Definition:** The preimage of an image is the set of all values in the codomain such that their mappings are all in a specified image. This image may be the codomain or some other set.
+
+T^-1(S) = Preimage of S under T.
+
+This can also be stated as T'(S)
+
+## Finding
+
+To find the preimage of some image under T we need to find all input vectors a such that T(a) is in the image.
+
+If we specify the image as <1,2> and <0,0> then we need to find all <x_1,x_2> such that <x_1, x_2> x L.T. Matrix = <1,2> or <0,0>.
+
+This final result can be found using [[ReducedRowEchelonForm.md]] of both augmented matricies created using the above information where the result is the computed values of all pivot variables put into matricies.
diff --git a/StatisticsAndProbability.md b/StatisticsAndProbability.md
@@ -80,4 +80,5 @@ L8:
- [[BernoulliRandomVariable.md]]
L9:
- [[JointDensityFunction.md]]
+L10:
- [[]]
