JSFHMath 2270-3kernel and image - big exampleSeptember 18, 2009 - extension of September 16 notesConsider T(x)=Ax, for the large matrixLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRLkxpbmVhckFsZ2VicmFGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiOkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVA==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JSFH1a) Find an explicit representation for ker(T) (which we also write as ker(A)). Use this representation to find a basis for ker(T) and verify that it's a basis. You might want to use: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVE2UmVkdWNlZFJvd0VjaGVsb25Gb3JtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEiQUYnRi9GMi9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSI7RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==JSFH1b) As it turns out, Maple has commands to find a basis for the kernel too....the synonym for kernel is nullspace. This word makes clear sense, since "null" means zero, and we are looking for domain vectors that are transformed to zero.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZFLUkjbWlHRiQ2JVEqTnVsbFNwYWNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEiQUYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEiO0YnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy1GQzYtUSJ+RidGQEZGL0ZJRj9GSkZMRk5GUEZSRlQvRlhGVkZaLUZDNi1RIiNGJ0ZARkZGZ25GSkZMRk5GUEZSRlRGaG4tRiw2JVFhb2lmfndlJ3Jlfmx1Y2t5LH5pdCdzfnRoZX5zYW1lfmFzfnRoZX5iYXNpc353ZX5mb3VuZH4tfml0fm5lZWR+bm90fmJlLH5GJ0YvRjItSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGVi8lJmRlcHRoR0Zkby8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0ZaRlpGWkZaRlpGWkZaRlpGWkZaRlpGWkZaRlpGWkZaRlpGWkZaRlpGWkZaRlpGaW4tRiw2JVFGfnNpbmNlfnN1YnNwYWNlfmJhc2VzfmFyZX5ub3R+dW5pcXVlIUYnRi9GMkY9RkA=2a) For the same transformation, find an explicit representation for Im(T). Use the fewest number of vectors which still span this subspace. Verify that they are a basis. Hint: reuse your work in the previous part to cull vectors from this "column space." You can read this information 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 column dependencies correspond to solutions to Ax=0, which do not change as you do elementary row operations. For example, since the third column of the reduced matrix is the sum of the first two, this must also be true for the original matrix!2b) A clever way to get an especially nice basis for Im(T) is to use "elementary column operations" to replace the column vectors (which span Im(T)) with a nicer collection which still spans. This amounts to computing the reduced column echelon form of the matrix: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) Let's compare our different basis choices for the Image to what Maple comes up with. The image of T is the span of the columns of A, and this subspace is also known by the synonym "column space".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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=2d) What space to you think Maple is finding a basis for with this command, and is it a subset of the domain or the target?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2e) How is the Row space of A related to the kernel of A?JSFHTTdSMApJNFJUQUJMRV9TQVZFLzE5NDM0ODhYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISM5IiUiJyIiIiIiISIiIyIiJEYpRichIiUhIiNGKkYnRixGJyEiIkYsRi1GLEYsRi0iIiVGKSIiJ0YqRioiIio2Ig==TTdSMApJNFJUQUJMRV9TQVZFLzIyNjQ2MzZYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISM5IiUiJyIiIiIiIUYoRihGKEYnRihGKEYnRidGKEYoRihGKEYnRihGKCEiIkYoRigiIiRGJ0YpRig2Ig==TTdSMApJNFJUQUJMRV9TQVZFLzIyNjQ4MjRYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISInIichIiQhIiIiIiEiIiJGKUYqNiI=TTdSMApJNFJUQUJMRV9TQVZFLzIzNDc1MjBYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISInIiciIiEiIiJGJ0YnRihGJzYiTTdSMApJNFJUQUJMRV9TQVZFLzIzNDc1OTJYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISInIichIiJGJyIiIiIiIUYpRik2Ig==TTdSMApJNFJUQUJMRV9TQVZFLzIzNDc2ODhYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISM5IiUiJyIiIiIiIUYoRihGKEYnRihGKEYnRidGKEYoRihGKEYnRihGKCEiIkYoRigiIiRGJ0YpRig2Ig==TTdSMApJNFJUQUJMRV9TQVZFLzIzNDc4OTJYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISM5IiciJSIiIiIiIyIiJCEiIiEiIyIiJyIiIUYnRidGK0YqRilGKCEiJUYrRioiIiVGKUYpRitGJ0YrRigiIio2Ig==TTdSMApJNFJUQUJMRV9TQVZFLzIzNDgwOTZYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISM5IiUiJyIiIiIiIUYoRidGKEYnRihGKEYoRihGJ0YnRihGKEYoRihGKEYoRihGKEYoRihGKEYoNiI=TTdSMApJNFJUQUJMRV9TQVZFLzIzNDgzMDBYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIlIiUiIiIiIiFGKEYnNiI=TTdSMApJNFJUQUJMRV9TQVZFLzIzNjQ4MTZYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIlIiUiIiEiIiJGJ0YnNiI=TTdSMApJNFJUQUJMRV9TQVZFLzIzNjQ4ODhYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIlIiUiIiFGJyIiIkYoNiI=TTdSMApJNFJUQUJMRV9TQVZFLzIzODEzOTJYKiUpYW55dGhpbmdHNiI2IltnbCEkJSEhISInIiciIiIiIiFGJ0YoRigiIiQ2Ig==TTdSMApJNFJUQUJMRV9TQVZFLzIzODE2MzJYKiUpYW55dGhpbmdHNiI2IltnbCEkJSEhISInIiciIiEiIiJGKEYnISIiRig2Ig==TTdSMApJNFJUQUJMRV9TQVZFLzIzODE3MDRYKiUpYW55dGhpbmdHNiI2IltnbCEkJSEhISInIiciIiFGJ0YnIiIiRichIiI2Ig==